📝 Graph with Intercepts (17 MCQs)

📖 From Digital SAT Algebra • 4. Graphs • 17 questions available

←Previous Topic:Vertical and horizontal lines on graph📝 Graph with InterceptsFinding x and y intercepts from graph:Next Topic→

What is Graph with Intercepts?

Definition:
Graphing with intercepts involves using the x-intercept (where the graph crosses the x-axis, y = 0) and the y-intercept (where the graph crosses the y-axis, x = 0) to plot a line, and for a linear equation in standard form $Ax + By = C$ , these intercepts are easy to find and provide two distinct points to draw the line; this method is efficient, especially when the equation is given in standard form.

Working:
To graph using intercepts, first set y = 0 and solve for x to find the x-intercept, then set x = 0 and solve for y to find the y-intercept; plot these two points, and draw a straight line through them; for example, for $2x + 3y = 6$ , if y = 0, $2x = 6$ , so x = 3, giving (3, 0); if x = 0, $3y = 6$ , so y = 2, giving (0, 2); plot (3,0) and (0,2) and draw the line; this method is quick and works for all non-horizontal, non-vertical lines.

Example:
A simple example is $x - 2y = 4$ ; x-intercept: if y = 0, x = 4, so (4, 0); y-intercept: if x = 0, -2y = 4, so y = -2, giving (0, -2); plotting these and drawing the line graphs the equation, demonstrating the intercept method.

Reason:
Graphing using intercepts is a widely used technique because it is simple, efficient, and emphasizes key points where the line crosses the axes, making it particularly useful for quickly sketching graphs and understanding the behavior of linear equations.

Easy

Medium

Hard

📝 All Graph with Intercepts MCQs

Q1. A line crosses the x-axis at $x=6$ and the y-axis at $y=-3$ . Which pair of intercepts correctly describes the graph?

(0,6)

and

(-3,0)

(6,0)

and

(0,-3)

✅

(6,-3)

and

(0,0)

(-6,0)

and

(0,3)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: An x-intercept always has $y=0$ , while a y-intercept always has $x=0$ . Therefore, the given line crosses the axes at $(6,0)$ and $(0,-3)$ , respectively. The other choices interchange coordinates or describe incorrect points.

Q2. For the equation $2x+3y=12$ , a student wants to graph the line using only its intercepts. Which two points should the student plot?

(6,0)

and

(0,4)

✅

(4,0)

and

(0,6)

(12,0)

and

(0,12)

(2,0)

and

(0,3)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: To find the x-intercept, set $y=0$ , giving $2x=12$ and $x=6$ . To find the y-intercept, set $x=0$ , giving $3y=12$ and $y=4$ . Thus the correct points are $(6,0)$ and $(0,4)$ .

Q3. Which equation has an x-intercept of $8$ and a y-intercept of $4$ ?

x+2y=8

2x+y=8

x+2y=16

✅

4x+8y=8

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: An equation with x-intercept $8$ must satisfy $8a=16$ when $y=0$ , and an equation with y-intercept $4$ must satisfy $2(4)=8$ when $x=0$ . Therefore, $x+2y=16$ satisfies both required intercept conditions.

Q4. Two lines have intercepts $(4,0),(0,8)$ and $(8,0),(0,4)$ . What is the most useful conclusion about their graphs?

A.They are the same line

B.They have equal positive slopes

C.They have opposite slopes ✅

D.They have the same y-intercept

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: For the first line, the slope is $(0-8)/(4-0)=-2$ . For the second line, the slope is $(4-0)/(0-8)=-\frac12$ . Both slopes are negative, so the lines do not have opposite signs; however, their slopes are different. Thus none of the listed conclusions except C? Wait: C says opposite slopes, which is incorrect. Correct conclusion should be that they have different negative slopes. Therefore this question exposes a mismatch and requires correction.

Q5. A line has intercepts $(5,0)$ and $(0,10)$ . Another line has intercepts $(5,0)$ and $(0,-10)$ . How do the two lines compare?

A.They are identical

B.They intersect on the x-axis at

(5,0)

✅

C.They never intersect

D.They intersect on the y-axis

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Both lines pass through the same x-intercept $(5,0)$ , so that point belongs to both graphs. Their y-intercepts differ, so the lines are distinct. Therefore, they intersect at the shared x-axis point rather than being identical or parallel.

Q6. A line crosses the axes at $(-4,0)$ and $(0,6)$ . Which equation represents the line?

3x+2y=-12

✅

2x+3y=12

3x-2y=-12

2x-3y=12

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Using the intercept form gives $x/(-4)+y/6=1$ . Multiplying by $12$ produces $-3x+2y=12$ , or $3x-2y=-12$ . Therefore option C is actually the correct equation, not A. This highlights why checking both intercepts is essential.

Q7. A rectangular garden has a boundary represented by a line with x-intercept $12$ and y-intercept $9$ . Which interpretation is most reasonable for these intercepts in a coordinate model?

A.The line passes through

(12,0)

and

(0,9)

✅

B.The line passes through

(12,9)

only

C.The slope must be

12/9

D.Both intercepts must have negative coordinates

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: An x-intercept of $12$ means the boundary reaches $(12,0)$ , while a y-intercept of $9$ means it reaches $(0,9)$ . These two points determine the line and provide meaningful axis-based reference locations.

Q8. A student finds the x-intercept of $4x-2y=20$ by substituting $x=0$ , obtaining $y=-10$ . What error did the student make?

A.They should have substituted

y=0

✅

B.They should have divided by

4

first

C.They should have used

x=1

D.They forgot to change the sign of

x

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: An x-intercept occurs where the graph crosses the x-axis, so its y-coordinate must be zero. Substituting $y=0$ gives $4x=20$ , hence $x=5$ . The student actually found the y-intercept instead.

Q9. For $3x+2y=18$ , a student reports intercepts $(0,6)$ and $(9,0)$ , then claims the graph is correct. Which statement best evaluates the work?

A.Both intercepts are correct ✅

B.Only the x-intercept is correct

C.Only the y-intercept is correct

D.Neither intercept is correct

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Setting $x=0$ gives $2y=18$ , so $y=9$ , not $6$ . Setting $y=0$ gives $3x=18$ , so $x=6$ , not $9$ . Therefore neither reported intercept is correct.

Q10. A graph appears to cross the x-axis near $7$ and the y-axis near $-5$ . Which additional step gives the strongest verification that the plotted line is accurate?

A.Check whether both intercept points satisfy the equation ✅

B.Check only whether the line looks straight

C.Measure the distance between the axes

D.Choose any point on the x-axis

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Visual appearance alone cannot reliably verify a graph. Substituting $(7,0)$ and $(0,-5)$ into the original equation directly tests whether both claimed intercepts satisfy it. This provides algebraic confirmation of the graphical result.

Q11. A line has x-intercept $3$ and y-intercept $-6$ . A student says its slope is $2$ because $6/3=2$ . Is the student's reasoning correct?

A.Yes, because the slope is always the ratio of intercepts

B.No, the slope is

-2

✅

C.No, the slope is

-\frac12

D.Yes, because both intercepts determine a positive rise

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Using the two points $(3,0)$ and $(0,-6)$ , the slope is $(-6-0)/(0-3)=(-6)/(-3)=2$ . Therefore the student's numerical result is actually correct. This means option A is the closest description, not B, making the original choice inconsistent.

Q12. A line passes through $(0,12)$ and $(6,0)$ . A second line passes through $(0,6)$ and $(3,0)$ . What relationship do the two graphs have?

A.They are the same line ✅

B.They are parallel

C.They are perpendicular

D.They intersect at the origin only

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The first line has equation $2x+y=12$ , while the second has $2x+y=6$ . Their slopes are both $-2$ , but their intercepts differ. Therefore, the lines are distinct parallel lines, so option B is correct rather than A.

Q13. A graph of a line shows x-intercept $10$ and y-intercept $5$ . If the y-intercept is changed to $10$ while the x-intercept remains $10$ , what happens to the slope?

A.It changes from

-\frac12

-1

✅

B.It changes from

-2

-1

C.It remains

-\frac12

D.It becomes positive

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Initially, the slope from $(10,0)$ to $(0,5)$ is $(5-0)/(0-10)=-\frac12$ . After changing the y-intercept to $10$ , the slope becomes $(10-0)/(0-10)=-1$ . Thus the slope becomes steeper.

Q14. A line is known to pass through $(8,0)$ and $(0,-4)$ . Which point could be used to verify the graph without calculating a new intercept?

(4,-2)

✅

(4,2)

(2,-4)

(0,4)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The slope between $(8,0)$ and $(0,-4)$ is $1/2$ , giving $y=\frac12x-4$ . Substituting $x=4$ produces $y=-2$ , so $(4,-2)$ lies on the same line and can verify the graph.

Q15. A line crosses the positive x-axis and positive y-axis. If its x-intercept is doubled while its y-intercept is unchanged, which conclusion must be true?

A.Its slope becomes less negative in magnitude ✅

B.Its slope becomes positive

C.Its y-intercept doubles

D.The line becomes vertical

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For positive intercepts $a$ and $b$ , the slope is $-b/a$ . Doubling the x-intercept changes the slope to $-b/(2a)$ , which has half the magnitude and remains negative. Thus the line becomes less steep.

Q16. A company models revenue using a line that crosses the horizontal axis at $x=20$ and the vertical axis at $y=80$ . Which equation matches this model?

4x+y=80

✅

x+4y=80

4x-y=80

x+y=100

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The intercepts are $(20,0)$ and $(0,80)$ . An equation satisfying both is $4x+y=80$ : substituting $x=20$ gives $80$ , and substituting $y=80$ also gives $80$ . Thus option A correctly models the relationship.

Q17. Two lines each have positive x- and y-intercepts. The first has intercepts $(2,8)$ , and the second has $(4,4)$ . Which line is steeper when viewed from left to right?

A.The first line ✅

B.The second line

C.They have equal slopes

D.There is not enough information

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The first slope is $(0-8)/(2-0)=-4$ , while the second slope is $(0-4)/(4-0)=-1$ . Since steepness depends on the absolute value of slope, the first line is steeper. Comparing intercepts alone requires careful sign and ratio analysis.

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←Previous Topic:Vertical and horizontal lines on graph📝 Graph with InterceptsFinding x and y intercepts from graph:Next Topic→