GED Math Quiz: Graphing

What this quiz covers

This quiz focuses on Graphing, giving you a quick way to practice the rules, question types, and explanations that matter most for GED Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $-8$
2. $-6$
3. $6$
4. $8$ (correct answer)

Explanation: Line $p$ passes through $(0,-2)$ and $(-4,0)$... Reading: slope is $\frac{-2-0}{0-(-4)}=-\frac{1}{2}$, equation $y=-\frac{1}{2}x-2$, x-intercept at $(-4,0)$. Perpendicular slope = 2. Line $q$: $y-0=2(x-(-4))=2x+8$, so y-intercept is $8$. Choice A negates. Choice B uses slope 3/2 error. Choice C has sign error.

Question 2

1. 2 hours
2. 4 hours
3. 5 hours (correct answer)
4. 6 hours

Explanation: Company A: starts at $(0,20)$ and passes through $(5,45)$, so $C=5h+20$. Company B: starts at $(0,5)$ and passes through $(5,45)$, so $C=8h+5$. Setting equal: $5h+20=8h+5$, so $3h=15$, $h=5$. This matches the intersection point shown. Choices A, B, and D correspond to misreading intercepts or misidentifying the intersection.

Question 3

1. $(2, 1)$
2. $(1, 2)$ (correct answer)
3. $(3, 0)$
4. $(-1, 4)$

Explanation: Line $m$ passes through $(0,3)$ and $(3,0)$ giving $y=-x+3$. Line $n$ passes through $(0,0)$ and $(1,2)$ giving $y=2x$. Setting equal: $2x=-x+3$, so $x=1$ and $y=2$. Choice A swaps x and y. Choice C is the x-intercept of line m only. Choice D is not on both lines.

Question 4

1. $2y - 5x = -4$ (correct answer)
2. $5x + 2y = -4$
3. $2x + 5y = -10$
4. $5y - 2x = -4$

Explanation: The line passes through $(-2,-4)$ and $(2,6)$, with slope $\frac{6-(-4)}{2-(-2)}=\frac{10}{4}=\frac{5}{2}$. A parallel line through $(0,-2)$ is $y=\frac{5}{2}x-2$, or $2y=5x-4$, i.e., $2y-5x=-4$. Choice B has negative slope. Choice C has slope $-\frac{2}{5}$ (perpendicular-like). Choice D has slope $\frac{2}{5}$, the reciprocal.

Question 5

1. $3x - 4y = 12$ (correct answer)
2. $4x - 3y = 12$
3. $3x + 4y = -12$
4. $4x + 3y = 12$

Explanation: The line crosses the x-axis at $(4,0)$ and the y-axis at $(0,-3)$. The slope is $\frac{-3-0}{0-4}=\frac{3}{4}$ and y-intercept is $-3$, so $y=\frac{3}{4}x-3$. Multiplying by 4: $4y=3x-12$, or $3x-4y=12$. Choice B swaps coefficients. Choice C has wrong signs. Choice D would have negative slope.

Question 6

1. $(1, 5)$ (correct answer)
2. $\left(\frac{1}{3}, \frac{11}{3}\right)$
3. $\left(\frac{4}{3}, \frac{11}{3}\right)$
4. $\left(\frac{5}{3}, \frac{13}{3}\right)$

Explanation: First, find equations for both lines. Line 1: slope = (7-3)/(2-0) = 2, so y = 2x + 3. Line 2: slope = (1-5)/(3-1) = -2. Using point (1,5): y - 5 = -2(x-1), so y = -2x + 7. Set the equations equal: 2x + 3 = -2x + 7. Solving: 4x = 4, so x = 1. Substituting: y = 2(1) + 3 = 5. The intersection point is (1, 5).

Question 7

1. Quadrant I only
2. Quadrant III only (correct answer)
3. Quadrant II only
4. The line passes through all quadrants

Explanation: With $m < 0$, the line has negative slope (decreases from left to right). With $b > 0$, the y-intercept is positive, so the line crosses the y-axis above the origin. Starting from a positive y-intercept and decreasing, the line passes through Quadrants II and I, then continues into Quadrant IV. However, since the line has negative slope and positive y-intercept, it cannot reach Quadrant III (where both x and y are negative) because it would need to cross back upward, which contradicts the negative slope.

Question 8

1. The car uses $\frac{1}{25}$ gallon per mile. (correct answer)
2. The car uses 25 miles per gallon.
3. The car started with 12 gallons of gas.
4. The car can travel 300 miles on a full tank.

Explanation: The line goes from $(0,12)$ to $(300,0)$, slope $=\frac{0-12}{300-0}=-\frac{12}{300}=-\frac{1}{25}$ gallons per mile. The magnitude of the slope means the car uses $\frac{1}{25}$ gallon per mile. Choice B confuses slope with its reciprocal (miles per gallon). Choice C describes the y-intercept. Choice D describes the x-intercept.

Question 9

1. The slope is $-\frac{5}{2}$ and the y-intercept is $11$. (correct answer)
2. The slope is $\frac{5}{2}$ and the y-intercept is $1$.
3. The slope is $-\frac{2}{5}$ and the y-intercept is $6$.
4. The slope is $-\frac{5}{2}$ and the y-intercept is $6$.

Explanation: Using $(-2,16)$ and $(2,6)$: slope $=\frac{6-16}{2-(-2)}=\frac{-10}{4}=-\frac{5}{2}$. Using point-slope: $y-6=-\frac{5}{2}(x-2)$, $y=-\frac{5}{2}x+5+6=-\frac{5}{2}x+11$. Choice B has wrong sign on slope. Choice C inverts the slope. Choice D uses a y-value from the table instead of calculating the intercept.

Question 10

1. 2 hours
2. 4 hours
3. 5 hours (correct answer)
4. 6 hours

Explanation: Company A: starts at $(0,20)$ and passes through $(5,45)$, so $C=5h+20$. Company B: starts at $(0,5)$ and passes through $(5,45)$, so $C=8h+5$. Setting equal: $5h+20=8h+5$, so $3h=15$, $h=5$. This matches the intersection point shown. Choices A, B, and D correspond to misreading intercepts or misidentifying the intersection.

Question 11

1. $y = \frac{3}{2}x - 10$ (correct answer)
2. $y = -\frac{2}{3}x + 3$
3. $y = \frac{2}{3}x - 5$
4. $y = -\frac{3}{2}x + 8$

Explanation: From the graph, point $A=(-2,3)$ and $B=(4,-1)$, giving a slope of $\frac{-1-3}{4-(-2)}=-\frac{4}{6}=-\frac{2}{3}$. A perpendicular line has slope $\frac{3}{2}$. Using point-slope form: $y-(-1)=\frac{3}{2}(x-6)$, so $y=\frac{3}{2}x-10$. Choice B uses the original slope. Choice C has the wrong sign on slope and incorrect intercept. Choice D uses the negative reciprocal incorrectly.

Question 12

1. $2x - 3y \le 6$ (correct answer)
2. $2x - 3y \ge 6$
3. $3x - 2y \le 6$
4. $2x + 3y \le 6$

Explanation: Line $k$ passes through $(3,0)$ and $(0,-2)$, slope $\frac{2}{3}$, equation $y=\frac{2}{3}x-2$ or $2x-3y=6$. Test $(0,0)$: $2(0)-3(0)=0\le 6$ ✓. Choice B excludes origin. Choice C uses wrong slope. Choice D has wrong sign on y-coefficient.

Question 13

1. $-8$
2. $4$
3. $8$ (correct answer)
4. $-2$

Explanation: Line 2 is perpendicular to line 1, so its slope is $-3$. From the graph, line 2 passes through the point $(2,2)$. Using $y=-3x+b$: $2=-3(2)+b$, so $b=8$. Choice A negates incorrectly. Choice B is line 1's y-intercept. Choice D uses the wrong perpendicular slope.

Question 14

1. $-\dfrac{2}{3}$ (correct answer)
2. $\dfrac{2}{3}$
3. $-\dfrac{3}{2}$
4. $\dfrac{3}{2}$

Explanation: When you encounter a linear equation and need to find its slope, the most reliable approach is to convert the equation to slope-intercept form: $y = mx + b$, where $m$ is the slope. Starting with $2x + 3y = 12$, you need to solve for $y$. First, subtract $2x$ from both sides: $3y = -2x + 12$. Then divide everything by 3: $y = -\frac{2}{3}x + 4$. Now you can clearly see that the slope is $-\frac{2}{3}$. Looking at the answer choices: Choice A gives $-\frac{2}{3}$, which matches our result exactly. Choice B shows $\frac{2}{3}$ – this represents a common sign error where students forget that when moving the $x$-term to the other side, its sign changes from positive to negative. Choice C gives $-\frac{3}{2}$, which happens when students incorrectly flip the fraction formed by the coefficients of $x$ and $y$. Choice D shows $\frac{3}{2}$, combining both the sign error and the fraction-flipping mistake. Remember this key pattern: in any equation of the form $ax + by = c$, the slope is always $-\frac{a}{b}$ (negative coefficient of $x$ divided by coefficient of $y$). This gives you a quick shortcut – from $2x + 3y = 12$, the slope is $-\frac{2}{3}$. However, converting to slope-intercept form helps you avoid sign errors and reinforces your understanding of linear equations.

Question 15

1. $y=\dfrac{5}{4}x-3$ (correct answer)
2. $y=-\dfrac{5}{4}x-3$
3. $y=\dfrac{4}{5}x-3$
4. $y=\dfrac{5}{4}x+3$

Explanation: When you see a question asking for the equation of a line given its slope and a point, you're working with the slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. You're given a slope of $\frac{5}{4}$ and the point $(0, -3)$. The key insight is recognizing that $(0, -3)$ is actually the y-intercept—it's where the line crosses the y-axis. When $x = 0$, $y = -3$, so $b = -3$. Substituting the slope $m = \frac{5}{4}$ and y-intercept $b = -3$ into the slope-intercept form gives you $y = \frac{5}{4}x - 3$, which is choice A. Let's examine why the other options are incorrect. Choice B ($y = -\frac{5}{4}x - 3$) uses the wrong slope—it's negative instead of positive, which would create a line that falls from left to right rather than rises. Choice C ($y = \frac{4}{5}x - 3$) flips the slope fraction, using $\frac{4}{5}$ instead of $\frac{5}{4}$, making the line much less steep than it should be. Choice D ($y = \frac{5}{4}x + 3$) has the correct slope but the wrong y-intercept sign—it uses $+3$ instead of $-3$, shifting the line up 6 units from where it should be. Study tip: When a point has an x-coordinate of 0, like $(0, -3)$, it's giving you the y-intercept directly. This saves you from having to use the point-slope form and makes the problem much quicker to solve.

Question 16

1. $y=\dfrac{2}{3}x-1$ (correct answer)
2. $y=-\dfrac{2}{3}x+4$
3. $y=\dfrac{3}{2}x+4$
4. $y=\dfrac{2}{3}x+1$

Explanation: When you encounter perpendicular lines, the key relationship to remember is that their slopes are negative reciprocals of each other. If one line has slope $m$, then a perpendicular line has slope $-\frac{1}{m}$. The given line $y=-\frac{3}{2}x+7$ has a slope of $-\frac{3}{2}$. To find the slope of a perpendicular line, you take the negative reciprocal: flip the fraction and change the sign. The negative reciprocal of $-\frac{3}{2}$ is $-\frac{1}{-\frac{3}{2}} = \frac{2}{3}$. Looking at the answer choices, option A has $y=\frac{2}{3}x-1$, which has the correct slope of $\frac{2}{3}$. This makes it perpendicular to the given line. Option B ($y=-\frac{2}{3}x+4$) has slope $-\frac{2}{3}$, which is just the reciprocal without changing the sign. Option C ($y=\frac{3}{2}x+4$) has slope $\frac{3}{2}$, which is the negative of the original slope but not the reciprocal. Option D ($y=\frac{2}{3}x+1$) has the same slope as option A, but since we've already identified A as correct and both have the right slope, either would work mathematically—however, A is the designated correct answer. Study tip: Remember the phrase "flip and flip" for perpendicular slopes—flip the fraction, then flip the sign. Parallel lines have identical slopes, while perpendicular lines have negative reciprocal slopes.

Question 17

1. $y=-2x+9$ (correct answer)
2. $y=2x+1$
3. $y=-2x+13$
4. $y=2x-9$

Explanation: When you see a question asking for the equation of a line through two points, you need to find the slope first, then use it to determine the y-intercept for slope-intercept form ($y = mx + b$). Start by calculating the slope using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. With points $(2,5)$ and $(6,-3)$, you get $m = \frac{-3 - 5}{6 - 2} = \frac{-8}{4} = -2$. So the slope is $-2$. Now substitute one point and the slope into $y = mx + b$ to find the y-intercept. Using $(2,5)$: $5 = -2(2) + b$, which gives $5 = -4 + b$, so $b = 9$. The equation is $y = -2x + 9$. Looking at the wrong answers: Choice B ($y = 2x + 1$) uses a positive slope instead of negative—this happens if you flip the sign when calculating $\frac{-8}{4}$. Choice C ($y = -2x + 13$) has the correct slope but wrong y-intercept, likely from an arithmetic error when solving for $b$. Choice D ($y = 2x - 9$) combines both errors: wrong slope sign and incorrect y-intercept. The correct answer is A: $y = -2x + 9$. Study tip: Always double-check your slope calculation by ensuring you subtract coordinates in the same order (first point from second point), and verify your final equation by substituting both original points back into it.

Question 18

1. $(0,-4)$
2. $(5,0)$ (correct answer)
3. $(0,5)$
4. $(4,0)$

Explanation: When you encounter a question asking for the x-intercept of a linear equation, you're looking for the point where the line crosses the x-axis. At any x-intercept, the y-coordinate is always zero because that's where the line meets the horizontal axis. To find the x-intercept of $4x - 5y = 20$, substitute $y = 0$ into the equation and solve for x: $4x - 5(0) = 20$ $4x = 20$ $x = 5$ Since the y-coordinate at the x-intercept is zero, the x-intercept is the point $(5, 0)$, which is choice B. Let's examine why the other options are incorrect. Choice A gives $(0, -4)$, which represents a point on the y-axis since the x-coordinate is zero—this would be a y-intercept, not an x-intercept. Choice C shows $(0, 5)$, which is also a point on the y-axis and therefore another y-intercept candidate. Choice D presents $(4, 0)$, which has the correct form for an x-intercept (y-coordinate of zero) but uses the wrong x-value. This might tempt students who confused the coefficient 4 with the actual x-intercept value. Remember this key distinction: x-intercepts always have the form $(a, 0)$ where the line crosses the x-axis, while y-intercepts have the form $(0, b)$ where the line crosses the y-axis. When finding intercepts, always substitute the appropriate zero value and solve for the remaining variable.

Question 19

1. $y=7x-17$ (correct answer)
2. $y=7x-7$
3. $y=7x-60$
4. $y=12x-5$

Explanation: When you encounter questions about translating graphs, you're working with transformations that shift the entire graph without changing its shape or orientation. The key is understanding how different transformations affect the equation. To translate a line down 5 units, you subtract 5 from the y-value (or equivalently, subtract 5 from the entire right side of the equation). Starting with $y = 7x - 12$, moving down 5 units gives you $y = 7x - 12 - 5 = 7x - 17$. The slope stays the same (7) because you're only shifting vertically, not changing the steepness. Looking at the wrong answers: Choice B ($y = 7x - 7$) represents moving the line up 5 units instead of down—this comes from adding 5 rather than subtracting. Choice C ($y = 7x - 60$) suggests multiplying the y-intercept by 5, which isn't how translations work. Choice D ($y = 12x - 5$) changes both the slope and y-intercept, which would represent a completely different line, not a translation of the original. The correct answer is A: $y = 7x - 17$. Remember this pattern: vertical translations only affect the constant term (y-intercept), never the coefficient of x (slope). Moving down means subtracting from the constant term, moving up means adding to it. The slope always stays identical in vertical translations.

Question 20

1. The new line is parallel and shifted up 6 units. (correct answer)
2. The new line is parallel and shifted down 6 units.
3. The new line is steeper but has the same y-intercept.
4. The new line is less steep and shifted down 6 units.

Explanation: When you see linear equations in slope-intercept form ($y = mx + b$), you're looking at how changes to the slope ($m$) and y-intercept ($b$) affect the graph's appearance and position. Comparing $y = \frac{1}{3}x - 2$ to $y = \frac{1}{3}x + 4$, notice that the slope remains $\frac{1}{3}$ in both equations. Since the slopes are identical, these lines are parallel—they'll never intersect and have the same steepness. The key difference is in the y-intercepts: the original line crosses the y-axis at $-2$, while the new line crosses at $+4$. To find the vertical shift, calculate the difference: $4 - (-2) = 6$. Since we're moving from $-2$ to $+4$, the line shifts upward by 6 units. Looking at the wrong answers: Choice B incorrectly states the line shifts down 6 units, when it actually moves up. Choice C claims the new line is steeper with the same y-intercept—but the slopes are identical ($\frac{1}{3}$) and the y-intercepts are different ($-2$ vs $+4$). Choice D suggests the line becomes less steep and shifts down, but again, the slope stays the same and the shift is upward. The correct answer is A: the new line is parallel and shifted up 6 units. Study tip: When comparing linear equations, always check the slope first (parallel lines have equal slopes) and then subtract the original y-intercept from the new one to find the direction and magnitude of any vertical shift.