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MK · lesson 14 of 14 · 42 min

Coordinate Geometry and Graphs

Coordinate geometry links algebra and shapes, and the Mathematics Knowledge subtest uses it to test both at once. This advanced lesson starts on the coordinate plane itself: positive and negative directions, the four quadrants, reading graph scales, reflections and translations. It then covers horizontal, vertical and general distance, the midpoint formula and finding a missing endpoint.

Reading · 42 minVideo · 14 minPractice questions (49)

Video lesson: Coordinate Geometry and Graphs

Video transcript

0:00 Welcome to the explainer. Today, we are going to absolutely crush coordinate geometry for the ASVAB Mathematics Knowledge Subtest. And honestly, the best part, we're going to do the whole thing without ever touching a calculator. Since the ASVAB doesn't allow calculators anyway, the test makers really aren't looking for complex, messy math. They just want to see if you can use quick, careful reasoning. So by the end of our time today, you'll have all the visual tools you need to spot the easy shortcuts and grab those points for sure. Alright, here's our game plan for today. We're going to build this step by step. First up, the grid and moving points. Then we'll tackle distance and midpoint formulas, followed by slopes and line equations. After that, we'll look at lines, shapes, and curves. We'll wrap up the lesson part with some shortcuts and common traps to avoid. And finally, a check yourself practice section to see how much you've learned. Let's get right into it. The grid and moving points. This is the absolute foundation. You know, the actual board where all this math is taking place. Now, what's kind of amazing is just how simple the coordinate plane actually is. It's literally just two number lines crossing at a right angle.

1:01 You've got your horizontal line, the x-axis, and your vertical line, the y-axis. And they meet right in the middle at the origin, 0-0. These axes split the whole board into four quadrants, and we number them counterclockwise. Notice how this works. In quadrant one, the upper right, both x and y are positive. Move over to the upper left, quadrant two, x is negative, but y is still positive. Down to the lower left, quadrant three, both are negative. And finally, the lower right, quadrant four, x is positive, but y is negative. Honestly, drawing a super quick rough sketch of these positive and negative directions on your scratch paper is going to be your absolute best friend on test day. Let's put this into practice with our first worked example. We're going to see how to flip and slide a point logically. So, we want to take the point three, negative two, reflect it across the y-axis, and then move it six units up. Okay, step one is the reflection. Think of the y-axis like a giant mirror. When we reflect across it, the y-value stays exactly the same, but the x-value flips its sign. So, positive three becomes negative three. Our new point is negative three, negative two.

2:05 Simple, right? Now, step two is a translation, which really just means sliding the point around. Moving up means we add to the y-coordinate. So, we take our y-value of negative two, add six to it, and boom, we get positive four. So, our final resting point is negative three, comma four, beautifully simple. Distance and midpoint formulas. Now that we can confidently place and move single points around, let's figure out how to measure the space between two of them. So, the midpoint is exactly halfway between two endpoints. Normally, you just average the x's and average the y's and call it a day. But what if the test gives you the midpoint and asks you to find the missing endpoint? Look at step one here. The midpoint M is one five. One endpoint A is negative three two. We need to find the other endpoint B. The brilliant shortcut here is to just work backward. You simply double the midpoint and subtract the known endpoint. Let's look at step two for the x-coordinate. We double the midpoint's x-value of one, which gives us two, then we subtract negative three. Now, two minus negative three is exactly like saying two plus three, right?

3:08 Which is five. On to step three for the y-coordinate. Double the midpoint's y-value of five, giving us ten. Subtract two, which leaves us with eight. Put it all together in step four, and our missing endpoint B is exactly five eight. Okay, so what about the actual distance between two points? If points line up horizontally or vertically, it's super easy. You just subtract. But when they're diagonal, like going from negative four one all the way to four sixteen, well, we use the distance formula. And here's a crucial little secret. The distance formula is really just the Pythagorean theorem in disguise. Step one. Find the horizontal and vertical changes. Going from negative four to positive four is a change of eight. From one to sixteen is a change of fifteen. Step two, we square those changes and add them up. Eight squared is sixty-four, and fifteen squared is two hundred twenty-five. Add it together, we get two hundred eighty-nine. Finally, step three, we take the square root of two hundred eighty-nine, which is seventeen.

4:08 Even better, if you memorize your classic right triangle patterns, like three, four, five, or five, twelve, thirteen, or in the specific case, eight, fifteen, seventeen, you can completely skip that square root step. Total time saver. All right, let's keep this momentum going with section three. Slopes and line equations. We know how to handle points, so now let's explore what happens when we connect them to form continuous lines. Slope basically measures steepness, right? The classic rise over run. But listen, you really have to watch out for a classic trap the testmakers absolutely love to throw you. You must know the difference between a zero slope and an undefined slope. A horizontal line, like y equals four, is perfectly flat. It goes zero units up for every step it takes to the right, and zero divided by anything is just zero, so a horizontal line has a slope of exactly zero. On the flip side, a vertical line, like x equals five, goes straight up and down. It has literally zero horizontal movement, and in math, dividing by zero is totally impossible. Therefore, the slope of a vertical line is undefined.

5:12 Just remember, undefined is definitely not the same as zero. Don't let them trick you on that. Let's build an equation completely from scratch now. Say a line passes through the points, zero, negative three, and two, five. First up, we need the slope, which we call m. We divide the change in y by the change in x. In step two, we calculate that rise over run. Five minus negative three on top, which becomes five plus three, or eight. On the bottom, two minus zero is just two. Eight divided by two gives us a nice clean slope of four. Step three is finding our equation in that familiar y equals mx plus b form, where b is our y intercept. Let's look back at our points. The point zero negative three actually tells us exactly where the line crosses the y axis, simply because x is zero, so our y intercept is negative three. We just plug our slope and intercept right into the formula, giving us y equals four x minus three. Now sometimes equations are written in standard form, something like five x plus two y equals 20.

6:12 Finding the intercepts here is just brilliantly simple. You literally just cover up one variable. Seriously, step one. To find the x intercept, we set y to zero. Imagine covering up that two y part with your finger. You're left with five x equals 20. Divide by five and x is four, so our x intercept is four zero. Step two, do the same for x, set x to zero, cover up the five x, and you have two y equals 20. Divide by two and y is 10. Our y intercept is zero 10. Once you have those two points, plotting the line is an absolute breeze. And as step three shows us, if you ever need the slope straight from standard form, the quick trick is negative a over b. So in this case, that's negative five over two. Lines, shapes, and curves. Let's combine all these tools we've learned to look at multiple lines crossing and basic geometric shapes sitting directly on our grid. Remembering exactly how line slopes relate to each other will let you instantly identify their relationships without ever having to draw a graph, which is huge for saving time.

7:14 Parallel lines run side by side forever. They never ever meet. That means they must rise and run at the exact same rate. So parallel lines always have equal slopes. Perpendicular lines are a totally different story. They crash right into each other to form perfect right angles. Their slopes are what we call negative reciprocals. Basically, you take the fraction, flip it upside down, and change the sign from positive to negative, or vice versa. The cool thing is, when you multiply two perpendicular slopes together, they will always equal exactly negative one. Let's see this in action with a worked example. Step one, we're given the equation three x minus y equals two. Before we do anything else, we really should rewrite it into y equals mx plus b form, just so we can see the slope clearly. Moving things around gives us y equals three x minus two. Okay, perfect. So our slope is three. Step two, a parallel line needs the exact same slope, so its slope is also three. Easy. Step three, for a perpendicular line, we take that number three, flip it to one third and make it negative.

8:22 So the perpendicular slope is negative one third, and step four proves it for us. Three times negative one third equals negative one. See, it all connects. But what if lines aren't parallel? Well, that means they cross, and when lines intersect, that exact meeting point makes both equations true simultaneously. Look at step one. Where do y equals two x plus one, and y equals negative x plus ten, meet. Since both of these equations equal y, step two is to literally just set them equal to each other. So two x plus one equals negative x plus ten. We add x to both sides, subtract one from both sides, and we get three x equals nine. Divide by three, and we find that x equals three. Now for step three. Plug that three back into either of the original equations. Let's just use the first one. Two times three is six, plus one is seven. So y equals seven. Step four gives us our final intersection point. Three comma seven. The ASVAB is also going to place shapes right on the coordinate grid. Let's look at a rectangle with corners at negative two, negative three, six negative three, six two, and negative two two. Because the sides of this rectangle align perfectly with the grid lines, finding the dimensions is literally just simple subtraction.

9:33 Step two, for the width, just subtract the matching horizontal x coordinates. Six minus negative two is eight. For the height, subtract the vertical y coordinates. Two minus negative three is five. Now that we know the rectangle is eight by five, the rest is basic geometry you probably already know. Step three, the perimeter is the distance all the way around. So two times the sum of eight plus five equals 26 units. Step four, the area is just width times height. So eight times five, which is 40 square units. And this next example brilliantly illustrates how we can use the distance formula to test points on a circle. Step one, a circle has its center at one negative two and radius of thirteen. The question asks, is the point six ten sitting right on the edge of that circle? Well, if it is, the distance from the center to that point must equal exactly thirteen. Let's test it. Step two, find the change in x and y. From one to six is a change of five. From negative two to ten is a change of twelve. Moving to step three, we have five squared plus twelve squared. Wait a second. That's our familiar five twelve thirteen right triangle pattern.

10:38 Since five squared plus twelve squared equals thirteen squared, the distance is exactly thirteen. Step four, yes, it perfectly lies on the circle. And look at that, no calculator required. We are on the home stretch now folks. Section five, shortcuts and common traps. We've built a really strong foundation today, so let's quickly spotlight the specific traps the test makers want you to fall into so you can easily just step right over them. I want you to take a mental snapshot of these rules right now. These are the most common coordinate traps on the mathematics knowledge subtest. First up, swapping x and y. Always, always, always list the horizontal x first. Second, reflecting the wrong coordinate. Remember our mirror analogy? If you reflect across the x axis, it's the y value that changes its sign. Third, mixing your subtraction order when finding slope. If you start with the second point on top, you absolutely must start with the second point on the bottom. You gotta keep it consistent. Fourth, calling a vertical slope zero.

11:38 Remember, vertical slopes are undefined. Fifth, looking at standard form, like 4x plus 2y equals 12, and thinking the slope is just four. It's not. You have to rearrange it for y first. And finally, making a slope a fraction for a perpendicular line, but forgetting to make it negative. If you avoid these mistakes and sketch out your problems quickly, you are going to save so many points. Okay, section six, check yourself practice. You have all the visual tools you've got the mental shortcuts. Let's test your confidence with two real practice questions straight from the material. The point five negative three is reflected across the x axis. What are the coordinates of its image? Think back to our very first examples. Picture that mirror. What changes when we jump across the horizontal x axis? I'll give you a second to think. Alright, the correct answer is A, five three. Exactly. When you reflect across the x axis, the x value stays exactly where it is. Only the y value flips its sign so that negative three becomes positive three.

12:41 The new point is five three. Nailed it. Let's look at question two. What is the midpoint of the segment joining the point four negative two and the point ten eight for our final challenge today? How do we find the exact center between these two points? Take a moment to solve it. And the correct answer is B, seven three. You simply average the coordinates. Add the x's together, four plus ten is fourteen, divided by two is seven. Add the y's together, negative two plus eight is positive six, divided by two is three. Giving you a midpoint of seven three. This proves that with careful steps, you can conquer any coordinate geometry question on the ASVAB without ever reaching for a calculator. So what other mathematical mysteries could you unlock just by learning the visual rules of the game? Thanks for joining me on this explainer and keep up the great practice.

Contents · 48 sections
  1. Lesson Goal and ASVAB Context
  2. The Coordinate Plane
  3. Positive and Negative Directions
  4. The Four Quadrants
  5. Reading Graph Scales Carefully
  6. Reflections Across the Axes
  7. Translations in the Plane
  8. Horizontal and Vertical Distance
  9. General Distance Formula
  10. Choosing Efficient Distance Methods
  11. The Midpoint Formula
  12. Finding an Endpoint From a Midpoint
  13. Slope as Rise Over Run
  14. Positive, Negative, Zero and Undefined Slope
  15. Slope From Two Points
  16. Slope From a Graph
  17. Slope-Intercept Form
  18. Interpreting the Y-Intercept
  19. Interpreting the X-Intercept
  20. Converting Standard Form
  21. Graphing a Line From Two Points
  22. Finding a Line Equation From Slope and Intercept
  23. Point-Slope Form
  24. Finding an Equation From Two Points
  25. Parallel Lines
  26. Perpendicular Lines
  27. Intersections as Solutions
  28. Geometric Meaning of Systems
  29. Symmetry and Coordinates
  30. Rectangles on the Coordinate Plane
  31. Right Triangles From Coordinates
  32. Distance to a Circle Center
  33. Reading Nonlinear Graphs Without Assuming a Line
  34. Graphs of Inequalities: Boundary Only
  35. Sketch Before Doing Algebra
  36. No-Calculator Coordinate Shortcuts
  37. Answer-Choice Elimination
  38. Worked Example 1 — Reflection and Translation
  39. Worked Example 2 — Midpoint and Unknown Endpoint
  40. Worked Example 3 — Pythagorean Distance
  41. Worked Example 4 — Slope and Intercept
  42. Worked Example 5 — Standard Form and Two Intercepts
  43. Worked Example 6 — Parallel and Perpendicular
  44. Worked Example 7 — Intersection of Two Lines
  45. Worked Example 8 — Rectangle Perimeter and Area
  46. Worked Example 9 — Circle Coordinate Test
  47. Common MK Coordinate Traps
  48. Final Mathematics Knowledge Review
Visual lesson

Lesson slides

15 slides
Coordinate Geometry and Graphs – slide 1 of 15

Lesson Goal and ASVAB Context

By the end of this lesson you should be able to place and read points on a grid, name the quadrant or axis a point sits on, and move points by reflecting or sliding them. You will also find distances, midpoints, slopes, and intercepts, write and graph line equations, recognize parallel and perpendicular lines, and locate where two lines cross.

These skills support the Mathematics Knowledge (MK) part of the ASVAB, where short coordinate problems reward quick, careful reasoning more than long computation.

The Coordinate Plane

A coordinate plane is built from two number lines that cross at right angles. The horizontal line is the x-axis, the vertical line is the y-axis, and they meet at the origin, (0, 0).

Every point is named by an ordered pair (x, y). The first number tells how far to move left or right from the origin; the second tells how far to move up or down. Order matters: (2, 5) and (5, 2) are different points.

Coordinate plane with the x-axis, y-axis and origin (0, 0); the points (2, 5) and (5, 2) are plotted in different places.

Positive and Negative Directions

On the x-axis, positive values lie to the right of the origin and negative values to the left. On the y-axis, positive values lie above the origin and negative values below. So (-3, -1) means three units left and one unit down.

The Four Quadrants

The axes split the plane into four quadrants, numbered counterclockwise from the upper right. Quadrant I has signs (+, +), II has (-, +), III has (-, -), and IV has (+, -). A point with a zero coordinate, such as (0, -6), lies on an axis and belongs to no quadrant.

The four quadrants numbered counterclockwise: I (+, +), II (−, +), III (−, −), IV (+, −); the point (0, −6) lies on the y-axis, not in a quadrant.

Reading Graph Scales Carefully

Grid lines do not always stand for one unit. Before reading a point, check the labels on each axis. If every grid line on the x-axis marks 5 units, a point three lines to the right has x = 15, not 3. The two axes may even use different scales, so read each one separately.

Grid whose x-axis lines are labelled 0, 5, 10, 15, 20 and y-axis lines 0, 1, 2, 3; a point three grid lines to the right has x = 15, not 3.

Reflections Across the Axes

A reflection flips a point over a line like a mirror. Reflecting across the x-axis keeps x and changes the sign of y: (x, y) becomes (x, -y). Reflecting across the y-axis changes the sign of x instead: (x, y) becomes (-x, y). For example, (2, -6) reflected across the x-axis is (2, 6).

The point (2, −6) reflected across the x-axis lands at (2, 6): x stays 2 and y changes sign.

Translations in the Plane

A translation slides a point without flipping or turning it. Moving right adds to x, moving left subtracts from x, moving up adds to y, and moving down subtracts from y. Shifting (1, 4) three units right and five units down gives

(1 + 3, 4 - 5) = (4, -1)
The point (1, 4) slides 3 units right and 5 units down to (4, −1).

Horizontal and Vertical Distance

When two points share a y-value, they sit on a horizontal line, and the distance is the difference of the x-values. From (-2, 3) to (6, 3) is

6 - (-2) = 8

units. When they share an x-value, subtract the y-values: (5, -4) to (5, 2) is 6 units. Distance is never negative.

Horizontal segment from (−2, 3) to (6, 3) is 8 units long; vertical segment from (5, −4) to (5, 2) is 6 units long.

General Distance Formula

For points that do not line up, use the distance formula, which comes from the Pythagorean theorem:

From (-1, 2) to (2, 6), the changes are 3 and 4, so

Points (−1, 2) and (2, 6) joined by a right triangle with legs 3 and 4; the distance between them is the hypotenuse, 5.

Choosing Efficient Distance Methods

First check whether the points share a coordinate; if so, simply subtract. If not, find the horizontal and vertical changes and see whether they match a familiar right-triangle pattern such as 3-4-5, 6-8-10, 5-12-13, or 8-15-17. From (0, 0) to (5, 12) the legs are 5 and 12, so the distance is 13 without any square-root work.

The Midpoint Formula

The midpoint is the point exactly halfway between two endpoints. Average the x-values and average the y-values:

For (2, -6) and (10, 4), the midpoint is (6, -1).

Segment from (2, −6) to (10, 4) with its midpoint M (6, −1) marking two equal halves.

Finding an Endpoint From a Midpoint

If you know the midpoint M and one endpoint A, the other endpoint is

B = 2M - A

coordinate by coordinate. With M = (3, 1) and

A = (-1, 4)
B = (6 + 1, 2 - 4) = (7, -2)

Slope as Rise Over Run

Slope measures steepness: how much a line goes up or down for each step to the right.

Rise is the vertical change and run is the horizontal change. A ramp that climbs 2 feet over a horizontal stretch of 8 feet has slope

Ramp drawn as a right triangle with run 8 ft and rise 2 ft, so slope = 2/8 = 1/4.

Positive, Negative, Zero and Undefined Slope

Reading left to right, a line with positive slope climbs and a line with negative slope falls. A horizontal line, such as y = 4, has a rise of 0, so its slope is 0. A vertical line, such as x = 5, has a run of 0; division by zero is impossible, so its slope is undefined, which is not the same as zero.

Left: a rising line has positive slope and a falling line has negative slope. Right: the horizontal line y = 4 has slope 0 and the vertical line x = 5 has undefined slope.

Slope From Two Points

Given two points, divide the change in y by the change in x:

For (1, -2) and (4, 7):

Subtract in the same order on top and bottom.

Slope From a Graph

Pick two points where the line crosses grid corners exactly, then count. If a line passes through (0, 1) and (3, -1), it drops 2 units while moving 3 units right, so

Line through (0, 1) and (3, −1): a run of 3 to the right and a drop of 2 give slope −2/3.

Always check the axis scale before counting.

Slope-Intercept Form

The form y = mx + b shows two facts at once: m is the slope, and b is the y-intercept, where the line crosses the y-axis at (0, b). In

y = 4x - 5

the slope is 4 and the line crosses the y-axis at (0, -5). Solve for y first if the equation is arranged differently.

Line y = 4x − 5 crossing the y-axis at (0, −5), the y-intercept b = −5; a step of run 1 and rise 4 from that point shows slope m = 4.

Interpreting the Y-Intercept

The y-intercept is the output when the input is zero. If renting a pressure washer costs

C = 3h + 20 dollars

for h hours, the y-intercept 20 is the flat fee paid even before any time is used, and the slope 3 is the added cost per hour.

Graph of C = 3h + 20 with hours h from 0 to 5 and cost C in dollars from 0 to 40: the line starts at the flat fee $20 and rises $3 for each hour.

Interpreting the X-Intercept

The x-intercept is where a line crosses the x-axis, so y equals 0 there. To find it, set y = 0 and solve. For

y = 2x - 8
0 = 2x - 8

gives x = 4, so the x-intercept is (4, 0).

Converting Standard Form

Standard form is Ax + By = C. Solving for y gives a slope of . For

6x + 3y = 9

subtract 6x and divide by 3:

The slope is -2 and the y-intercept is (0, 3).

Graphing a Line From Two Points

Two points fix a line. For standard form, the intercepts are often the easiest pair. For

x + 2y = 8

setting y = 0 gives (8, 0), and setting x = 0 gives (0, 4). Plot both, draw a straight line through them, and check a third point if time allows.

Graph of x + 2y = 8 drawn through its intercepts (8, 0) and (0, 4).

Finding a Line Equation From Slope and Intercept

When you know the slope and y-intercept, substitute them directly into y = mx + b. A line with slope and y-intercept 6 is

Check: at x = 4,

y = -2 + 6 = 4

so (4, 4) is on the line.

Point-Slope Form

If you know a slope m and any point , use

For slope 3 through (1, -2):

y + 2 = 3(x - 1)

which simplifies to

y = 3x - 5

Finding an Equation From Two Points

Find the slope first, then the intercept. For (-1, 4) and (3, -4),

m = -8 / 4 = -2

Substitute (-1, 4) into

y = -2x + b
4 = 2 + b

so b = 2 and the line is

y = -2x + 2

Parallel Lines

Parallel lines never meet, so they must rise at the same rate: they have equal slopes and different y-intercepts. The lines

y = 5x + 1
and
y = 5x - 3

are parallel. If an equation is in standard form, rewrite it as y = mx + b before comparing slopes. Two vertical lines are also parallel.

Parallel lines y = 5x + 1 and y = 5x − 3 with y-intercepts (0, 1) and (0, −3); each line has the same step of 1 right and 5 up, so both have slope 5 (x and y scales differ).

Perpendicular Lines

Perpendicular lines meet at a right angle. Their slopes are negative reciprocals, so their product is -1:

A line with slope is perpendicular to any line with slope . The exception is a horizontal line and a vertical line, which are perpendicular even though a vertical slope is undefined.

Two lines through the origin with slopes 2/5 and −5/2 meeting at a right angle; 2/5 × (−5/2) = −1.

Intersections as Solutions

The point where two lines cross makes both equations true at once. For

y = 3x - 1
and
y = -x + 7

set the right sides equal:

3x - 1 = -x + 7

so x = 2. Then

y = 3(2) - 1 = 5

and the lines meet at (2, 5).

Lines y = 3x − 1 and y = −x + 7 crossing at the point (2, 5).

Geometric Meaning of Systems

A pair of linear equations can behave in three ways. Lines with different slopes cross once, giving exactly one solution. Lines with the same slope but different intercepts are parallel and never meet, so there is no solution. Two equations that describe the very same line overlap everywhere, giving infinitely many solutions.

Three pairs of lines: lines that cross once (one solution), parallel lines (no solution), and two equations on the same line (infinitely many solutions).

Symmetry and Coordinates

Symmetry describes how a figure matches itself after a flip. A figure symmetric about the y-axis contains (-x, y) whenever it contains (x, y). Symmetry about the origin pairs (x, y) with (-x, -y). Reflecting across the line y = x swaps the coordinates, so (2, 7) becomes (7, 2).

Reflection across the dashed line y = x: (2, 7) and (7, 2) are joined by a segment perpendicular to y = x and split into equal halves; the coordinates swap. Point (x, y) with its mirror (−x, y) across the y-axis and its partner (−x, −y) through the origin.

Rectangles on the Coordinate Plane

When a rectangle's sides follow the grid lines, its width is a difference of x-values and its height is a difference of y-values. For corners (-3, 1), (4, 1), (4, 6), and (-3, 6), the width is 7 and the height is 5, so the area is 35 and the perimeter is 24.

Rectangle with corners (−3, 1), (4, 1), (4, 6) and (−3, 6) on a grid; width 7 and height 5.

Right Triangles From Coordinates

If two vertices share a y-value and two share an x-value, the triangle has a horizontal leg and a vertical leg. For (-1, -2), (4, -2), and (4, 10), the legs are 5 and 12.

The hypotenuse is

Right triangle with vertices (−1, −2), (4, −2) and (4, 10); right angle at (4, −2), legs 5 and 12, hypotenuse 13.

Distance to a Circle Center

Every point on a circle is the same distance, the radius r, from the center (h, k):

For center (2, -1) and radius 5, the point (5, 3) has changes 3 and 4, so its distance is 5 and it lies on the circle.

Circle with center (2, −1) and radius r = 5 on an equal-scale grid; the point (5, 3) is on the circle, joined to the center by a hypotenuse of 5 over dashed legs 3 and 4.

Reading Nonlinear Graphs Without Assuming a Line

Not every graph is a straight line. On the curve

the points (1, 1), (2, 4), and (3, 9) show y rising by 3 and then by 5 for equal steps in x. Because the rate of change keeps changing, you cannot use one slope for the whole graph. Read values directly from the curve or its equation instead of extending a straight line through two points.

Curve y = x² through (1, 1), (2, 4) and (3, 9): steps of 1 in x give rises of +3 then +5, while a dashed straight line through (1, 1) and (2, 4) wrongly predicts 7 at x = 3.

Graphs of Inequalities: Boundary Only

The boundary of an inequality such as

y > 2x + 1

is the line

y = 2x + 1

but the line itself is only the edge, not the answer. Draw it solid for

or

and dashed for < or >. Then shade the side that works: testing (0, 0) gives 0 > 1, which is false, so shade the side away from the origin.

Dashed boundary line y = 2x + 1 with the side away from the origin shaded for y > 2x + 1; the test point (0, 0) gives 0 > 1, false.

Sketch Before Doing Algebra

A quick rough sketch takes a few seconds and catches many errors. Plot the given points approximately and ask: Should the slope be positive or negative? Which quadrant should the answer land in? About how long should the segment be? If your computed answer disagrees with the sketch, recheck your signs and subtraction before choosing.

No-Calculator Coordinate Shortcuts

Watch for Pythagorean triples to avoid square roots. For a midpoint, average one coordinate at a time. For slope, simplify the fraction before dividing. To check whether a point is on a line, plug it in rather than rewriting the equation. To find intercepts in standard form, cover one variable, set it to zero, and divide.

Answer-Choice Elimination

Many coordinate questions can be checked by substituting choices. To decide which point lies on

y = -3x + 4

test each one: (2, -2) works because

-6 + 4 = -2

You can also drop choices with the wrong sign, the wrong quadrant, or a slope pointing the wrong way. Elimination is a check, not a substitute for understanding.

Worked Example 1 — Reflection and Translation

Problem: Reflect the point (3, −2) across the y-axis, then move it 6 units up.

Step 1: Reflecting across the y-axis flips the point left to right. The x-coordinate changes sign; y stays the same.

Step 2: Moving up adds to the y-coordinate. Add 6 to y; x stays the same.

Step 3: Simplify.

−2 + 6 = 4

Answer: The final point is (−3, 4).

The point (3, −2) reflected across the y-axis to (−3, −2), then moved up 6 to (−3, 4).

Check: Undo the moves. Go down 6:

(−3, 4 − 6) = (−3, −2)

Reflect back: (3, −2), the starting point.

Worked Example 2 — Midpoint and Unknown Endpoint

Problem: The midpoint of segment AB is (1, 5), and

A = (−3, 2)

Find B.

Givens for Worked Example 2: A (−3, 2) and midpoint M (1, 5); the step +4 right, +3 up from A to M is repeated from M to reach the unknown endpoint B = ?, with tick marks showing equal halves.

Step 1: The midpoint is the average of the endpoints, so twice the midpoint equals A + B. That means B = 2 × midpoint − A, done one coordinate at a time.

Step 2: Find the x-coordinate of B. Subtracting −3 is the same as adding 3.

Step 3: Find the y-coordinate of B.

Answer: B = (5, 8).

Check: Average A and B. x:

(−3 + 5) ÷ 2 = 2 ÷ 2 = 1

y:

(2 + 8) ÷ 2 = 10 ÷ 2 = 5

That gives (1, 5), the given midpoint.

Worked Example 3 — Pythagorean Distance

Problem: Find the distance from (−4, 1) to (4, 16).

Givens for Worked Example 3: points (−4, 1) and (4, 16) joined by a segment marked d = ?, with dashed horizontal and vertical legs (change in x, change in y) meeting at a right angle.

Step 1: Find the horizontal change (x) and the vertical change (y). Subtracting −4 is the same as adding 4.

Step 2: These changes are the legs of a right triangle. The distance is the hypotenuse, so square each leg.

Step 3: Add the squares and take the square root.

17 × 17 = 289

Answer: The distance is 17 units.

Check: 8-15-17 is a known right triangle. Also 17 is more than the longer leg, 15, and less than

8 + 15 = 23

as it should be.

Worked Example 4 — Slope and Intercept

Problem: A line passes through (0, −3) and (2, 5). Find its equation.

Givens for Worked Example 4: a line through the points (0, −3) and (2, 5), with its equation marked y = ?.

Step 1: Find the slope: change in y divided by change in x. Subtracting −3 is the same as adding 3.

Step 2: Find the y-intercept. The point (0, −3) has x = 0, so it is where the line crosses the y-axis. So

b = −3

Step 3: Write y = mx + b with m = 4 and

b = −3

Answer: The line is

y = 4x − 3

Check: Put in the other point, x = 2:

4(2) − 3 = 8 − 3 = 5

That matches (2, 5).

Worked Example 5 — Standard Form and Two Intercepts

Problem: For the line

5x + 2y = 20

find both intercepts and the slope.

Step 1: The x-intercept is where the line crosses the x-axis, so y = 0 there. Set y = 0 and solve.

So the x-intercept is (4, 0).

Step 2: The y-intercept is where x = 0. Set x = 0 and solve.

So the y-intercept is (0, 10).

Step 3: Find the slope from the two intercepts, (0, 10) and (4, 0).

Answer: x-intercept (4, 0), y-intercept (0, 10), slope −5/2.

Result of Worked Example 5: line 5x + 2y = 20 through the intercepts (4, 0) and (0, 10); a step right 2, down 5 shows slope −5/2.

Check: Solve for y:

2y = −5x + 20
so
y = −5/2

x + 10. The slope is −5/2 and the y-intercept is 10.

Worked Example 6 — Parallel and Perpendicular

Problem: Find the slopes of lines parallel and perpendicular to

3x − y = 2

Step 1: Get y alone to read the slope. Subtract 3x from both sides.

Step 2: Multiply both sides by −1 to make y positive.

The number in front of x is the slope, so the slope is 3.

Step 3: Parallel lines have the same slope, so a parallel line has slope 3.

Step 4: Perpendicular lines have the negative reciprocal slope. Flip 3 (which is 3/1) to get 1/3, then change the sign.

Answer: Parallel slope 3; perpendicular slope −1/3.

Check: Perpendicular slopes multiply to −1.

Worked Example 7 — Intersection of Two Lines

Problem: Where do the lines

y = 2x + 1
and
y = −x + 10

meet?

Step 1: At the meeting point both lines have the same y, so set the right sides equal.

Step 2: Add x to both sides to gather the x terms on the left.

Step 3: Subtract 1 from both sides, then divide by 3.

Step 4: Put x = 3 into the first equation to find y.

Answer: The lines meet at (3, 7).

Lines y = 2x + 1 and y = −x + 10 crossing at (3, 7).

Check: Put x = 3 into the second equation:

−3 + 10 = 7

Both lines give y = 7.

Worked Example 8 — Rectangle Perimeter and Area

Problem: A rectangle has corners (−2, −3), (6, −3), (6, 2) and (−2, 2). Find its perimeter and area.

Givens for Worked Example 8: rectangle with corners (−2, −3), (6, −3), (6, 2) and (−2, 2) on a grid; the bottom side is marked width ? and the right side height ?.

Step 1: Find the width from two corners on the same horizontal side, (−2, −3) and (6, −3). Subtract the x-values; subtracting −2 means adding 2.

Step 2: Find the height from two corners on the same vertical side, (6, −3) and (6, 2). Subtract the y-values.

Step 3: Perimeter is twice the sum of width and height.

Step 4: Area is width × height.

Answer: The perimeter is 26 units and the area is 40 square units.

Check: Add the four sides one by one:

8 + 5 + 8 + 5 = 26

Worked Example 9 — Circle Coordinate Test

Problem: A circle has center (1, −2) and radius 13. Is the point (6, 10) on the circle?

Step 1: Find the horizontal and vertical changes from the center to the point. Subtracting −2 means adding 2.

Step 2: A point is on the circle if its distance from the center equals the radius. Use the Pythagorean theorem: compare the sum of the squared changes with 13².

Step 3: The two results are equal, so the distance is exactly 13.

Answer: Yes, (6, 10) lies on the circle.

Check: By contrast, test (9, 4). The changes are

9 − 1 = 8
and
4 − (−2) = 6

and 8² + 6² =

64 + 36 = 100

so the distance is √100 = 10. Since 10 is less than 13, that point is inside the circle.

Circle with center (1, −2) and radius 13; a right triangle with legs 5 and 12 shows (6, 10) is 13 away and on the circle, while (9, 4) is 10 away and inside.

Common MK Coordinate Traps

Swapping coordinates: (4, -1) and (-1, 4) are different points. Always list x first.

Reflecting the wrong coordinate: across the x-axis, only y changes sign; across the y-axis, only x changes sign.

Mixing subtraction order in slope: if you start with the second point on top, start with it on the bottom too.

Calling a vertical slope zero: a vertical line has an undefined slope; a horizontal line has slope 0.

Using the coefficient of x in standard form as the slope: in

4x + 2y = 12

the slope is -2, not 4.

Forgetting to make a perpendicular slope negative: the perpendicular of 3 is -1/3, not 1/3.

Final Mathematics Knowledge Review

Points are written (x, y): right or left first, then up or down. The quadrant follows from the signs, and a zero coordinate puts a point on an axis. Reflections change one sign; translations add or subtract.

Distance uses subtraction for aligned points and the Pythagorean theorem otherwise. The midpoint averages the coordinates. Slope is change in y over change in x; horizontal lines have slope 0 and vertical lines have undefined slope.

In y = mx + b, m is the slope and b is the y-intercept. Find intercepts by setting the other variable to zero. Parallel lines share a slope; perpendicular slopes multiply to -1. An intersection is a point that satisfies both equations. Sketch first, then compute and check.

Quick review

What are the signs of x and y in each of the four quadrants?
Numbered counterclockwise from upper right: I (+, +), II (-, +), III (-, -), IV (+, -). A point with a zero coordinate lies on an axis, in no quadrant.
How does reflecting a point across the x-axis or the y-axis change it?
Across the x-axis: (x, y) becomes (x, -y). Across the y-axis: (x, y) becomes (-x, y). Across the line y = x the coordinates swap.
Distance formula between two points?
, from the Pythagorean theorem. Look for triples like 3-4-5 or 5-12-13 to skip the square root.
Midpoint formula, and how to find a missing endpoint?
. If M and endpoint A are known, the other endpoint is , coordinate by coordinate.
Slope from two points?
, rise over run. Subtract in the same order on top and bottom.
What is the slope of a horizontal line versus a vertical line?
Horizontal (y = c): slope 0. Vertical (x = c): run is 0, so slope is undefined, not zero.
In y = mx + b, what do m and b represent?
m is the slope (rate of change); b is the y-intercept, the point (0, b) where the line crosses the y-axis, i.e. the value of y when x = 0.
How do slopes of parallel and perpendicular lines compare?
Parallel lines have equal slopes (different intercepts). Perpendicular slopes are negative reciprocals: , so slope 3 pairs with .
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About this lesson

Slope comes next, as rise over run, from two points and from a graph, including zero and undefined slope. You will write and read slope-intercept, point-slope and standard form, interpret both intercepts, graph lines, and compare parallel and perpendicular lines. Intersections are treated as solutions to systems, and later sections apply coordinates to rectangles, right triangles and circle centers, with cautions about nonlinear graphs and inequality boundaries. Nine worked examples and a list of common coordinate traps support the material, along with sketching and answer-elimination shortcuts.

Set aside around 42 minutes. MK is part of the AFQT, so these skills carry weight. When you reach the final review, move on to the lesson test.

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