Question

use the graph of ( f(x) = |x| ) to write an equation for each function whose graph is shown.
(a)
( y = square \times )
(b)
( y = square \times )

Explanation:

Part (a)

Step 1: Identify Transformations

The parent function is \( f(x) = |x| \). The vertex of the given graph is at \( (-2, 0) \), so there is a horizontal shift left by 2 units. Also, the graph appears to be reflected vertically (since it opens downward) and possibly stretched or compressed. Let's check the slope. From the vertex \( (-2, 0) \) to a point like \( (0, -2) \), the slope is \( \frac{-2 - 0}{0 - (-2)} = -1 \). So the equation should be \( y = -|x + 2| \). Wait, let's verify. When \( x = -2 \), \( y = 0 \), which matches the vertex. When \( x = 0 \), \( y = -|0 + 2| = -2 \), which matches the point \( (0, -2) \). When \( x = -4 \), \( y = -|-4 + 2| = -|-2| = -2 \)? Wait, no, the graph at \( x = -4 \) should be \( y = -(-4 + 2) \)? Wait, maybe I made a mistake. Wait, the parent function \( |x| \) has vertex at (0,0). The given graph has vertex at (-2, 0), so horizontal shift left 2: \( |x + 2| \). Then, the graph opens downward, so multiply by -1: \( -|x + 2| \). But let's check the y-intercept. At \( x = 0 \), \( y = -|0 + 2| = -2 \), which matches the graph (the graph crosses the y-axis at (0, -2)). So the equation is \( y = -|x + 2| \). Wait, but maybe there's a vertical shift? Wait, the vertex is at (-2, 0), so no vertical shift. So the equation is \( y = -|x + 2| \).

Step 2: Verify

Check the vertex: \( x = -2 \), \( y = -|-2 + 2| = 0 \), correct. Check another point: \( x = -3 \), \( y = -|-3 + 2| = -|-1| = -1 \)? Wait, no, the graph at \( x = -3 \) should be \( y = -(-3 + 2) \)? Wait, maybe the slope is -1, so the left side (for \( x < -2 \)) has slope 1? Wait, no, if the vertex is at (-2, 0), and the graph goes through (0, -2), then for \( x \geq -2 \), the slope is \( \frac{-2 - 0}{0 - (-2)} = -1 \), so the equation for \( x \geq -2 \) is \( y = -1(x + 2) \), and for \( x < -2 \), the slope is 1 (since it's a V-shape opening downward, the left side has positive slope). So the equation is \( y = -|x + 2| \), which can be written as \( y = -|x + 2| \).

Step 1: Identify Transformations

The parent function is \( f(x) = |x| \). The vertex of the given graph is at \( (2, -6) \)? Wait, no, looking at the graph, the vertex is at (2, -6)? Wait, the graph shows the vertex at (2, -6)? Wait, the second graph (part b) has the vertex at (2, -6)? Wait, the user's graph for (b) has the vertex at (2, -6)? Wait, let's look again. The graph for (b) has the vertex at (2, -6)? Wait, the y-axis has marks, and the vertex is at (2, -6)? Wait, no, the graph shows the vertex at (2, -6)? Wait, the x-axis has marks at -2, 0, 2, 4, 6, 8. The y-axis has marks at -6, -4, -2, 0, 2. The vertex is at (2, -6)? Wait, no, the graph has a V-shape with vertex at (2, -6)? Wait, when x=2, y=-6. Then, let's check the slope. From the vertex (2, -6) to a point like (0, -4), the slope is \( \frac{-4 - (-6)}{0 - 2} = \frac{2}{-2} = -1 \). From the vertex (2, -6) to (4, -4), the slope is \( \frac{-4 - (-6)}{4 - 2} = \frac{2}{2} = 1 \). So the graph is a V-shape with vertex at (2, -6), opening upward (since the slope to the right is positive and to the left is negative, so it opens upward). So the equation is \( y = |x - 2| - 6 \). Let's verify: when x=2, y=|0| -6 = -6, correct. When x=0, y=|0 - 2| -6 = 2 -6 = -4, correct (matches the point (0, -4)). When x=4, y=|4 - 2| -6 = 2 -6 = -4, correct. When x=8, y=|8 - 2| -6 = 6 -6 = 0, which matches the graph (at x=8, y=0). So the equation is \( y = |x - 2| - 6 \).

Step 2: Verify

Check vertex: x=2, y=|2-2| -6 = -6, correct. Check x=0: y=| -2 | -6 = 2 -6 = -4, correct. Check x=4: y=|2| -6 = 2 -6 = -4, correct. Check x=8: y=|6| -6 = 0, correct. So the equation is \( y = |x - 2| - 6 \).

Answer:

\( y = -|x + 2| \)

Part (b)