Question

which of the following is the graph of this absolute value function?
$y = \frac{4}{3}|x|$

Explanation:

Step1: Recall absolute value function form

The general form of an absolute value function is \( y = a|x| \), where \( a \) determines the slope and direction. For \( y=\frac{4}{3}|x| \), when \( x = 0 \), \( y = 0 \), so the vertex is at \( (0,0) \). When \( x = 3 \), \( y=\frac{4}{3}\times3 = 4 \), so the point \( (3,4) \) should be on the graph. When \( x = 1 \), \( y=\frac{4}{3}\times1=\frac{4}{3}\approx1.33 \)? Wait, no, wait: Wait, let's check the graphs. Wait, the third graph: when \( x = 2 \), let's see. Wait, no, let's check the points. Wait, the function \( y=\frac{4}{3}|x| \): for \( x = 3 \), \( y = 4 \); for \( x = 1 \), \( y=\frac{4}{3}\approx1.33 \)? No, wait, maybe I miscalculated. Wait, no, let's check the graphs. The first graph: at \( x = 3 \), \( y = 4 \)? Wait, the first graph's blue dot is at \( (3,4) \)? Wait, no, the first graph's grid: x=3, y=4? Wait, the first graph's y-axis: 1,2,3,4,5. The blue dot is at (3,4). Let's check the function: when \( x = 3 \), \( y=\frac{4}{3}\times3 = 4 \), which matches. Wait, but the other graphs: the second graph, at x=1, y=3? No, \( y=\frac{4}{3}\times1=\frac{4}{3}\approx1.33 \), not 3. The third graph: at x=2, y=4? \( y=\frac{4}{3}\times2=\frac{8}{3}\approx2.66 \), not 4. Wait, wait, maybe I misread the graphs. Wait, the function is \( y=\frac{4}{3}|x| \), so the slope of the right branch (x ≥ 0) is \( \frac{4}{3} \), meaning for every 3 units right, 4 units up. So from (0,0), moving 3 right (x=3) gives y=4. Let's check the graphs:

- First graph: vertex at (0,0), and at x=3, y=4 (the blue dot). The slope from (0,0) to (3,4) is \( \frac{4 - 0}{3 - 0}=\frac{4}{3} \), which matches \( y=\frac{4}{3}|x| \).

- Second graph: at x=1, y=3? \( \frac{4}{3}\times1\approx1.33 eq3 \), so no.

- Third graph: at x=2, y=4? \( \frac{4}{3}\times2\approx2.66 eq4 \), so no. Wait, maybe the third graph's x=2 is y=4? Wait, no, the third graph's blue dot is at (2,4)? Wait, the problem's third graph: x=2, y=4? Then \( y=\frac{4}{3}\times2=\frac{8}{3}\approx2.66 \), not 4. Wait, maybe I made a mistake. Wait, the function is \( y=\frac{4}{3}|x| \), so the key points: vertex (0,0), and for x>0, slope \( \frac{4}{3} \). So when x=3, y=4 (since \( \frac{4}{3}\times3 = 4 \)). So the graph with the point (3,4) on the right branch (and symmetric left branch) is the first graph? Wait, no, the first graph's left branch: at x=-3, y=4, which is symmetric. Wait, but let's check the other graphs. Wait, the second graph: at x=1, y=3? No. The third graph: at x=2, y=4? No. Wait, maybe the first graph is correct? Wait, no, wait the first graph's y-axis: the grid lines. Wait, the first graph: when x=3, y=4. Let's calculate \( y=\frac{4}{3}\times3 = 4 \), which matches. So the first graph? Wait, no, wait the problem's first graph: the blue dot is at (3,4), which matches \( x=3, y=4 \). The other graphs: the second graph's blue dot is at (1,3), which would mean \( y=3 \) when \( x=1 \), but \( \frac{4}{3}\times1=\frac{4}{3}\approx1.33 eq3 \). The third graph's blue dot is at (2,4), \( y=\frac{4}{3}\times2=\frac{8}{3}\approx2.66 eq4 \). So the first graph? Wait, but wait, the first graph's left branch: at x=-3, y=4, which is symmetric, so that's correct. So the graph with vertex at (0,0) and passing through (3,4) (and (-3,4)) is the first graph? Wait, no, the first graph's x-axis: from -2 to 4, y-axis 1-5. The blue dot is at (3,4). So that's correct.

Wait, maybe I messed up. Let's re-express the function. The parent function is \( y = |x| \), which has a slope of 1 for x > 0. The function \( y=\frac{4}{3}|x| \) is a vertical stretch by a factor of \( \frac{4}{3} \), so the slope for x > 0 is \( \frac{4}{3} \), meaning for each unit increase in x, y increases by \( \frac{4}{3} \). So at x=3, y=4 (since \( 3\times\frac{4}{3}=4 \)), at x=1, y=\( \frac{4}{3}\approx1.33 \), at x=2, y=\( \frac{8}{3}\approx2.66 \). Now, looking at the graphs:

- First graph: at x=3, y=4 (matches \( 3\times\frac{4}{3}=4 \)). The left branch: at x=-3, y=4 (symmetric), so that's correct.

- Second graph: at x=1, y=3 (does not match \( \frac{4}{3}\approx1.33 \)).

- Third graph: at x=2, y=4 (does not match \( \frac{8}{3}\approx2.66 \)).

So the correct graph is the first one (the leftmost graph) with vertex at (0,0) and passing through (3,4).

Step2: Confirm the vertex and slope

The absolute value function \( y=\frac{4}{3}|x| \) has its vertex at (0,0) because when x=0, y=0. The coefficient \( \frac{4}{3} \) is positive, so the graph opens upwards. For x > 0, the function is \( y=\frac{4}{3}x \), which has a slope of \( \frac{4}{3} \). This means that for every 3 units we move to the right (increase x by 3), we move up 4 units (increase y by 4). The point (3, 4) satisfies this (since \( \frac{4}{3}(3) = 4 \)), which is shown in the first graph. The other graphs do not have the correct slope or points that satisfy the function.

Answer:

The left - most graph (with vertex at (0,0) and the point (3,4) on the right - hand branch)