Question

match each equation to the correct graph. the ^2 means to the second power.
graph a
graph b
graph c
graph d
f(x) = 1/3(x + 1)^2 - 5
f(x) = 3(x + 1)^2 - 5
f(x) = (x - 1)^2 - 5
f(x) = (x + 1)^2 + 5
f(x) = (x + 1)^2 - 5
f(x) = (x + 5)^2 - 1

Explanation:

To solve the problem of matching each quadratic function (in vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the vertical stretch/compression and direction) to its graph, we analyze each equation:

1. Recall the Vertex Form Properties:

• Vertex: \((h, k)\) (the point where the parabola changes direction).
• Vertical Stretch/Compression: \(|a| > 1\) stretches the parabola (narrower), \(0 < |a| < 1\) compresses it (wider).
• Direction: If \(a > 0\), the parabola opens upward; if \(a < 0\), it opens downward.

2. Analyze Each Equation:

Let’s list the equations (interpreting ^2 as squaring) and their key features:

Equation	Vertex \((h, k)\)	\(a\) (stretch/compression)	Direction
\( f(x) = 3(x + 1)^2 - 5 \)	\((-1, -5)\)	\( 3 \) (stretch, \(	3	> 1\))	Upward
\( f(x) = (x - 1)^2 - 5 \)	\((1, -5)\)	\( 1 \) (standard width)	Upward
\( f(x) = (x + 1)^2 + 5 \)	\((-1, 5)\)	\( 1 \) (standard width)	Upward
\( f(x) = (x + 1)^2 - 5 \)	\((-1, -5)\)	\( 1 \) (standard width)	Upward
\( f(x) = (x + 5)^2 - 1 \)	\((-5, -1)\)	\( 1 \) (standard width)	Upward

3. Match to Graphs (Using Vertex and Stretch):

• Graph A: Likely matches \( f(x) = \frac{1}{3}(x + 1)^2 - 5 \) (compressed, vertex \((-1, -5)\), opens upward).
• Graph B: Likely matches \( f(x) = 3(x + 1)^2 - 5 \) (stretched, vertex \((-1, -5)\), opens upward).
• Graph C: Likely matches \( f(x) = (x + 5)^2 - 1 \) (vertex \((-5, -1)\), standard width, opens upward).
• Graph D: Likely matches \( f(x) = (x - 1)^2 - 5 \) (vertex \((1, -5)\), standard width, opens upward) or \( f(x) = (x + 1)^2 + 5 \) (vertex \((-1, 5)\), standard width, opens upward—check the graph’s vertex position).

4. Final Matches (Example for Clarity):

For a precise match, confirm the vertex and stretch:

• \( f(x) = \frac{1}{3}(x + 1)^2 - 5 \): Vertex \((-1, -5)\), wide (compressed) → Graph A (wider parabola at \((-1, -5)\)).
• \( f(x) = 3(x + 1)^2 - 5 \): Vertex \((-1, -5)\), narrow (stretched) → Graph B (narrower parabola at \((-1, -5)\)).
• \( f(x) = (x - 1)^2 - 5 \): Vertex \((1, -5)\) → Graph D (vertex at \(x = 1\)).
• \( f(x) = (x + 1)^2 + 5 \): Vertex \((-1, 5)\) → A graph with vertex above the x-axis.
• \( f(x) = (x + 1)^2 - 5 \): Vertex \((-1, -5)\), standard width → A graph with vertex \((-1, -5)\) and standard width.
• \( f(x) = (x + 5)^2 - 1 \): Vertex \((-5, -1)\) → Graph C (vertex at \(x = -5\)).

Example Answer (Partial, Based on Common Patterns):

• \( f(x) = \frac{1}{3}(x + 1)^2 - 5 \) → Graph A (compressed, vertex \((-1, -5)\)).
• \( f(x) = 3(x + 1)^2 - 5 \) → Graph B (stretched, vertex \((-1, -5)\)).
• \( f(x) = (x - 1)^2 - 5 \) → Graph D (vertex \((1, -5)\)).
• \( f(x) = (x + 5)^2 - 1 \) → Graph C (vertex \((-5, -1)\)).

To finalize, cross-verify the vertex position and stretch with the graph’s shape (width, vertex coordinates) for each equation.

Answer:

To solve the problem of matching each quadratic function (in vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the vertical stretch/compression and direction) to its graph, we analyze each equation:

1. Recall the Vertex Form Properties:

• Vertex: \((h, k)\) (the point where the parabola changes direction).
• Vertical Stretch/Compression: \(|a| > 1\) stretches the parabola (narrower), \(0 < |a| < 1\) compresses it (wider).
• Direction: If \(a > 0\), the parabola opens upward; if \(a < 0\), it opens downward.

2. Analyze Each Equation:

Let’s list the equations (interpreting ^2 as squaring) and their key features:

Equation	Vertex \((h, k)\)	\(a\) (stretch/compression)	Direction
\( f(x) = 3(x + 1)^2 - 5 \)	\((-1, -5)\)	\( 3 \) (stretch, \(	3	> 1\))	Upward
\( f(x) = (x - 1)^2 - 5 \)	\((1, -5)\)	\( 1 \) (standard width)	Upward
\( f(x) = (x + 1)^2 + 5 \)	\((-1, 5)\)	\( 1 \) (standard width)	Upward
\( f(x) = (x + 1)^2 - 5 \)	\((-1, -5)\)	\( 1 \) (standard width)	Upward
\( f(x) = (x + 5)^2 - 1 \)	\((-5, -1)\)	\( 1 \) (standard width)	Upward

3. Match to Graphs (Using Vertex and Stretch):

• Graph A: Likely matches \( f(x) = \frac{1}{3}(x + 1)^2 - 5 \) (compressed, vertex \((-1, -5)\), opens upward).
• Graph B: Likely matches \( f(x) = 3(x + 1)^2 - 5 \) (stretched, vertex \((-1, -5)\), opens upward).
• Graph C: Likely matches \( f(x) = (x + 5)^2 - 1 \) (vertex \((-5, -1)\), standard width, opens upward).
• Graph D: Likely matches \( f(x) = (x - 1)^2 - 5 \) (vertex \((1, -5)\), standard width, opens upward) or \( f(x) = (x + 1)^2 + 5 \) (vertex \((-1, 5)\), standard width, opens upward—check the graph’s vertex position).

4. Final Matches (Example for Clarity):

For a precise match, confirm the vertex and stretch:

• \( f(x) = \frac{1}{3}(x + 1)^2 - 5 \): Vertex \((-1, -5)\), wide (compressed) → Graph A (wider parabola at \((-1, -5)\)).
• \( f(x) = 3(x + 1)^2 - 5 \): Vertex \((-1, -5)\), narrow (stretched) → Graph B (narrower parabola at \((-1, -5)\)).
• \( f(x) = (x - 1)^2 - 5 \): Vertex \((1, -5)\) → Graph D (vertex at \(x = 1\)).
• \( f(x) = (x + 1)^2 + 5 \): Vertex \((-1, 5)\) → A graph with vertex above the x-axis.
• \( f(x) = (x + 1)^2 - 5 \): Vertex \((-1, -5)\), standard width → A graph with vertex \((-1, -5)\) and standard width.
• \( f(x) = (x + 5)^2 - 1 \): Vertex \((-5, -1)\) → Graph C (vertex at \(x = -5\)).

Example Answer (Partial, Based on Common Patterns):

• \( f(x) = \frac{1}{3}(x + 1)^2 - 5 \) → Graph A (compressed, vertex \((-1, -5)\)).
• \( f(x) = 3(x + 1)^2 - 5 \) → Graph B (stretched, vertex \((-1, -5)\)).
• \( f(x) = (x - 1)^2 - 5 \) → Graph D (vertex \((1, -5)\)).
• \( f(x) = (x + 5)^2 - 1 \) → Graph C (vertex \((-5, -1)\)).

To finalize, cross-verify the vertex position and stretch with the graph’s shape (width, vertex coordinates) for each equation.