Question

function a and function b are linear functions.
function a
(graph of function a: a line on a coordinate grid with x from -10 to 10 and y from -10 to 10, passing through (0, -2) and other points)
function b

x	y
-6	-9
-3	-4
6	11

select all the statements that are true.

• the y-intercept of function a is equal to the y-intercept of function b.
• the y-intercept of function a is greater than the y-intercept of function b.
• the y-value of function a when x = 3 is less than the y-value of function b when x = 3.
• the y-value of function a when x = 3 is greater than the y-value of function b when x = 3.

Explanation:

Step1: Find y-intercept of Function A

The y-intercept is the value of \( y \) when \( x = 0 \). From the graph of Function A, when \( x = 0 \), \( y=-2 \). So y-intercept of A is \( -2 \).

Step2: Find y-intercept of Function B

For a linear function \( y = mx + b \), we first find the slope \( m \). Using two points from Function B's table, say \( (-6, -9) \) and \( (-3, -4) \), slope \( m=\frac{-4 - (-9)}{-3 - (-6)}=\frac{5}{3} \). Then use point \( (-3, -4) \) to find \( b \): \( -4=\frac{5}{3}(-3)+b \), \( -4=-5 + b \), so \( b = 1 \). Wait, no, let's recalculate. Wait, maybe better to use another point. Wait, when \( x = 0 \), let's find the equation. Wait, slope between \( (-6, -9) \) and \( (6, 11) \): \( m=\frac{11 - (-9)}{6 - (-6)}=\frac{20}{12}=\frac{5}{3} \). Then using \( x=-6, y = -9 \): \( -9=\frac{5}{3}(-6)+b \), \( -9=-10 + b \), so \( b = 1 \). Wait, but earlier when \( x = 0 \), let's plug into \( y=\frac{5}{3}x + 1 \), when \( x = 0 \), \( y = 1 \). Wait, but that contradicts? Wait no, maybe I made a mistake. Wait, let's check with \( (-3, -4) \): \( y=\frac{5}{3}(-3)+1=-5 + 1=-4 \), correct. So y-intercept of B is \( 1 \)? Wait no, wait the graph of A: when \( x = 0 \), y is -2. Wait, maybe I misread the graph. Wait the graph of Function A: the line crosses the y-axis at (0, -2), so y-intercept is -2. For Function B, let's find the equation again. Wait, maybe the first calculation was wrong. Let's take points (-6, -9) and (-3, -4). The change in y is \( -4 - (-9)=5 \), change in x is \( -3 - (-6)=3 \), so slope \( m=\frac{5}{3} \). Then using point (-6, -9): \( y - (-9)=\frac{5}{3}(x - (-6)) \), \( y + 9=\frac{5}{3}(x + 6) \), \( y=\frac{5}{3}x + 10 - 9 \), \( y=\frac{5}{3}x + 1 \). So when \( x = 0 \), \( y = 1 \). Wait, but that means y-intercept of A is -2, y-intercept of B is 1. So the first statement "The y-intercept of Function A is equal to the y-intercept of Function B" is false. The second statement "The y-intercept of Function A is greater than the y-intercept of Function B" is false because -2 < 1. Now let's find y at x=3 for both functions.

Step3: Find y at x=3 for Function A

Function A: from the graph, it's a linear function. Let's find its slope. From (0, -2) to (1, 0) (since when x=1, y=0? Wait no, the line passes through (1, 0)? Wait the graph: when x=1, y=0? Wait no, the line goes through (0, -2) and (1, 0)? Wait slope is \( \frac{0 - (-2)}{1 - 0}=2 \). So equation of A: \( y = 2x - 2 \). Let's check: when x=0, y=-2, correct. When x=1, y=0, correct. When x=3, \( y = 2(3)-2 = 4 \).

Step4: Find y at x=3 for Function B

Function B: equation is \( y=\frac{5}{3}x + 1 \). When x=3, \( y=\frac{5}{3}(3)+1 = 5 + 1 = 6 \). So y-value of A at x=3 is 4, y-value of B at x=3 is 6. So 4 < 6, so "The y-value of Function A when x = 3 is less than the y-value of Function B when x = 3" is true. The other statement "The y-value of Function A when x = 3 is greater than the y-value of Function B when x = 3" is false. Wait, but the original check marks: maybe I misread the graph. Wait let's re-examine Function A's graph. The line passes through (0, -2) and (2, 2)? Wait no, the graph: when x=1, y=0? Wait the grid: each square is 1 unit. So from (0, -2), going up 2, right 1: (1, 0), then (2, 2), (3, 4), (4, 6), (5, 8), (6, 10). Wait, when x=6, y=10? Wait the graph shows at x=7, y=10? Wait no, the arrow is at x=8, y=10? Wait maybe my initial slope calculation was wrong. Wait the graph: when x=0, y=-2; when x=1, y=0; x=2, y=2; x=3, y=4; x=4, y=6; x=5, y=8; x=6, y=10? Wait no, the graph's line at x=6 is at y=10? Wait the grid: x-axis from -10 to 10, y-axis from -10 to 10. The blue line starts at (-4, -10) and goes to (7, 10)? Wait maybe the slope is 2. So equation y = 2x - 2. Then at x=3, y=4. For Function B: let's recalculate the equation. Let's take points (-6, -9) and (6, 11). Slope is (11 - (-9))/(6 - (-6))=20/12=5/3≈1.666. Then equation: y - (-9)=(5/3)(x - (-6)), y + 9=(5/3)x + 10, y=(5/3)x + 1. So at x=3, y=(5/3)*3 + 1=5 + 1=6. So y of A at x=3 is 4, y of B at x=3 is 6. So 4 < 6, so the statement "The y-value of Function A when x = 3 is less than the y-value of Function B when x = 3" is true. Now y-intercepts: A is -2, B is 1. So -2 < 1, so the first two statements about y-intercepts are false. Wait but the original check marks: maybe the user made a mistake, but we need to correct. Wait maybe I misread Function A's y-intercept. Wait the graph: the line crosses the y-axis at (0, -2), correct. Function B's y-intercept: when x=0, y=1. So y-intercept of A is -2, B is 1. So "The y-intercept of Function A is equal to the y-intercept of Function B" is false. "The y-intercept of Function A is greater than the y-intercept of Function B" is false (since -2 < 1). Now for x=3: A's y is 4, B's y is 6. So "The y-value of Function A when x = 3 is less than the y-value of Function B when x = 3" is true. The other x=3 statement is false. Wait but the options: let's list the statements:

1. The y-intercept of Function A is equal to the y-intercept of Function B. (False)
2. The y-intercept of Function A is greater than the y-intercept of Function B. (False)
3. The y-value of Function A when x = 3 is less than the y-value of Function B when x = 3. (True)
4. The y-value of Function A when x = 3 is greater than the y-value of Function B when x = 3. (False)

Wait but maybe I made a mistake in Function B's equation. Let's check with x=6, y=11: y=(5/3)*6 + 1=10 + 1=11, correct. So equation is correct. Function A: from graph, when x=3, y=4 (since slope 2, y=2x - 2, x=3: 6 - 2=4). So yes, 4 < 6. So the only true statement is the third one: "The y-value of Function A when x = 3 is less than the y-value of Function B when x = 3". Wait but the original check marks in the image: maybe the user's initial checks were wrong, but we need to find the correct true statements.

Answer:

The true statement is: "The y-value of Function A when x = 3 is less than the y-value of Function B when x = 3." (The other statements are false. So among the given options, only this one is true.)