Question

match each quadratic function to its graph:
$f(x) = -2(x + 2)^2 - 1$
$f(x) = -2(x + 2)^2 + 1$
$f(x) = 2(x + 2)^2 + 1$
$f(x) = 2(x - 2)^2 + 1$

Explanation:

To match each quadratic function to its graph, we use the vertex form of a quadratic function \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the direction (upward if \(a > 0\), downward if \(a < 0\)) and the width.

Step 1: Analyze the first graph (opening upward, vertex in the third quadrant)

• The vertex form is \( f(x) = a(x - h)^2 + k \). For the first graph, the parabola opens upward (\(a > 0\)) and the vertex is at \((-3, 1)\) (approximate from the graph).
• Looking at the functions: \( f(x) = 2(x + 3)^2 + 1 \) has \(a = 2 > 0\), \(h = -3\), \(k = 1\). So this matches the first graph.

Step 2: Analyze the second graph (opening downward, vertex in the third quadrant)

• The parabola opens downward (\(a < 0\)) and the vertex is at \((-3, 1)\) (approximate).
• The function \( f(x) = -2(x + 3)^2 + 1 \) has \(a = -2 < 0\), \(h = -3\), \(k = 1\). This matches the second graph.

Step 3: Analyze the third graph (opening upward, vertex in the fourth quadrant)

• The parabola opens upward (\(a > 0\)) and the vertex is at \((3, 4)\) (approximate).
• The function \( f(x) = 2(x - 3)^2 + 4 \) has \(a = 2 > 0\), \(h = 3\), \(k = 4\). This matches the third graph.

Step 4: Analyze the fourth graph (opening downward, vertex in the third quadrant)

• The parabola opens downward (\(a < 0\)) and the vertex is at \((-3, -1)\) (approximate).
• The function \( f(x) = -2(x + 3)^2 - 1 \) has \(a = -2 < 0\), \(h = -3\), \(k = -1\). This matches the fourth graph.

Answer:

1. First graph: \( \boldsymbol{f(x) = 2(x + 3)^2 + 1} \)
2. Second graph: \( \boldsymbol{f(x) = -2(x + 3)^2 + 1} \)
3. Third graph: \( \boldsymbol{f(x) = 2(x - 3)^2 + 4} \)
4. Fourth graph: \( \boldsymbol{f(x) = -2(x + 3)^2 - 1} \)