Question

match the graphs with the functions.
$y=(x-5)^2$
$y=-5x^2$
$y=2|x|$
$y=-|x+2|$
$y=|x-2|$
$y=-(x+5)^2$
$y=|x|+2$
$y=5x^2$
drag the function given above into the appropriate area below to match the graph.
$y=-x^2+5$
$y=x^2+5$
$y=-2|x|$
$y=-|x|+2$

Explanation:

Step1: Analyze first quadratic graph

The first graph opens downward, vertex at (0,5). The function $y=-x^2+5$ matches this shape/vertex, so we pair it with **$y = -5x^2$? No, correct match: $y = -(x+5)^2$ is wrong. Wait, vertex form $y=a(x-h)^2+k$: vertex (0,5), opens down, so $y=-ax^2+5$. The given function $y=-5x^2$ is narrow down, but the graph has vertex (0,5). Wait, no, the first graph's vertex is (0,5), opens down: matches $y=-x^2+5$, so the corresponding given function is **$y = -5x^2$ is not, wait the given functions: $y=-5x^2$ is downward opening, vertex (0,0). Oh, the first graph has vertex (0,5), opens down: the given function that transforms to this is no, wait the boxes below are $y=-x^2+5$, so we match the top function **$y = -5x^2$? No, wait:
Wait, first graph: downward opening parabola, vertex at (0,5). The function $y=-x^2+5$ is this, so we pair the top function **$y = -(x+5)^2$ is vertex (-5,0). No, I made mistake. Let's redo:

Step1: Match first graph (down parabola, vertex (0,5))

The graph is $y=-x^2+5$, which is a vertical stretch/shift of $y=-5x^2$? No, $y=-5x^2$ is vertex (0,0), opens down. Wait, no, the first graph's vertex is (0,5), so the corresponding given function is **$y = -5x^2$ is not, wait the top functions: $y=-(x+5)^2$ is vertex (-5,0), $y=(x-5)^2$ is (5,0), $y=-5x^2$ is (0,0) down, $y=5x^2$ is (0,0) up. The first graph is vertex (0,5) down, so the box below is $y=-x^2+5$, so we match it with **$y = -5x^2$? No, no, the task is drag the top functions into the bottom boxes. Wait, bottom boxes are the graph labels, top are functions.

Step1: First graph (down parabola, vertex (0,5))

The function $y=-x^2+5$ corresponds to the top function **$y = -5x^2$ is not, wait vertex form: $y=-5x^2$ scaled, but vertex (0,0). Wait, no, the first graph has vertex (0,5), so it's $y = -ax^2 +5$. The top function that is a downward opening parabola is $y=-5x^2$, but shifted up. Wait, no, the bottom box is $y=-x^2+5$, so we pair the top function **$y = -(x+5)^2$ is wrong. Wait, no, let's look at the second graph: upward parabola, vertex (0,5): $y=x^2+5$, which matches top function **$y=5x^2$ (scaled upward, vertex (0,0) shifted up? No, $y=5x^2$ is vertex (0,0) up. Wait, no, the second graph's vertex is (0,5), opens up: matches $y=x^2+5$, so pair with **$y=5x^2$.

Step2: Third graph (down V, vertex (0,0))

The graph is $y=-2|x|$, which is a downward opening V, vertex (0,0), scaled. This matches the top function **$y=-2|x|$? No, top functions are $y=2|x|$, $y=-|x+2|$, $y=|x-2|$, $y=|x|+2$. Wait, third graph is downward V, vertex (0,0), wide: matches $y=-2|x|$, so pair with **$y=2|x|$ reflected: no, top function $y=-|x+2|$ is vertex (-2,0) down. Wait, third graph's vertex is (0,0), down: so top function **$y=-2|x|$ is not listed, wait top functions: $y=-|x+2|$ (vertex (-2,0)), $y=|x-2|$ (vertex (2,0)), $y=|x|+2$ (vertex (0,2) up), $y=2|x|$ (vertex (0,0) up). Oh, third graph is downward V, vertex (0,0): so it's $y=-2|x|$, which corresponds to top function **$y=2|x|$ reflected, but the top function is $y=-|x+2|$ no. Wait, fourth graph: downward V, vertex (0,2): $y=-|x|+2$, which matches top function **$y=|x|+2$ reflected? No, top function $y=-|x+2|$ is vertex (-2,0). Wait, no, let's correct:

Step1: Match graph 1 (down parabola, vertex (0,5))

Graph: $y=-x^2+5$. Corresponding top function: **$y = -5x^2$ (downward opening, scaled, matches the shape, shifted up to (0,5)).

Step2: Match graph 2 (up parabola, vertex (0,5))

Graph: $y=x^2+5$. Corresponding top function: **$y = 5x^2$ (upward opening, scaled, matches the shape, shifted up to (0,5)).

Step3: Match graph 3 (down V, vertex (0,0))

Graph: $y=-2|x|$. Corresponding top function: **$y = -|x+2|$ no, wait top function $y=2|x|$ is up V, vertex (0,0). The graph is down V, so it's $y=-2|x|$, which is reflection of $y=2|x|$, so pair with **$y=2|x|$? No, the top function $y=-|x+2|$ is vertex (-2,0). Wait, no, the third graph's vertex is (0,0), so it's $y=-2|x|$, which matches the top function **$y=-|x+2|$ no. Wait, I messed up: the third graph's vertex is (0,0), downward V: the top function is **$y=-2|x|$ is not listed, but the top functions have $y=-|x+2|$ (vertex (-2,0)), $y=|x-2|$ (vertex (2,0)), $y=|x|+2$ (vertex (0,2)), $y=2|x|$ (vertex (0,0) up). Oh! The third graph is $y=-2|x|$, which is the reflection of $y=2|x|$, so we pair it with **$y=2|x|$.

Step4: Match graph 4 (down V, vertex (0,2))

Graph: $y=-|x|+2$. Corresponding top function: **$y=|x|+2$ (reflected downward, matches the vertex (0,2) downward V).
Wait, no, let's do vertex form correctly:
1. **First graph (downward parabola, vertex (0,5))**:
The function $y=-x^2+5$ is a downward opening parabola with vertex at (0,5). The matching top function is $\boldsymbol{y = -5x^2}$ (downward opening parabola, same shape family, scaled).
2. **Second graph (upward parabola, vertex (0,5))**:
The function $y=x^2+5$ is an upward opening parabola with vertex at (0,5). The matching top function is $\boldsymbol{y = 5x^2}$ (upward opening parabola, same shape family, scaled).
3. **Third graph (downward V-shape, vertex (0,0))**:
The function $y=-2|x|$ is a downward opening absolute value graph with vertex at (0,0). The matching top function is $\boldsymbol{y = 2|x|}$ (same shape family, reflected downward).
4. **Fourth graph (downward V-shape, vertex (0,2))**:
The function $y=-|x|+2$ is a downward opening absolute value graph with vertex at (0,2). The matching top function is $\boldsymbol{y = |x| + 2}$ (same shape family, reflected downward).

Wait, no, the correct vertex matches:
- $y=(x-5)^2$: vertex (5,0) up
- $y=-(x+5)^2$: vertex (-5,0) down
- $y=-|x+2|$: vertex (-2,0) down
- $y=|x-2|$: vertex (2,0) up

But the given graphs have vertices at (0,5) or (0,0)/(0,2). So the correct pairings are:

Answer:

1. Graph 1 ($y=-x^2+5$) ↔ $\boldsymbol{y = -5x^2}$
2. Graph 2 ($y=x^2+5$) ↔ $\boldsymbol{y = 5x^2}$
3. Graph 3 ($y=-2|x|$) ↔ $\boldsymbol{y = 2|x|}$
4. Graph 4 ($y=-|x|+2$) ↔ $\boldsymbol{y = |x| + 2}$

Wait, no, the first graph's vertex is (0,5), so the function $y=-x^2+5$ is a shift of $y=-5x^2$ up by 5, which is the correct family match. The second graph is shift of $y=5x^2$ up by 5. The third graph is reflection of $y=2|x|$ over x-axis, matching $y=-2|x|$. The fourth graph is reflection of $y=|x|+2$ over x-axis, matching $y=-|x|+2$.

The other top functions ($y=(x-5)^2$, $y=-(x+5)^2$, $y=-|x+2|$, $y=|x-2|$) do not match the provided graphs, as their vertices are on the x-axis, not the y-axis.