Question

using structure in exercises 27-30, describe the transformation of the graph of the parent quadratic function. then identify the vertex.

1. $f(x)=3(x + 2)^2 + 1$
2. $f(x)=-4(x + 1)^2 - 5$
3. $f(x)=-2x^2 + 5$
4. $f(x)=\frac{1}{2}(x - 1)^2$

Explanation:

Step1: Recall parent quadratic form

The parent quadratic function is $f(x)=x^2$, with vertex form $f(x)=a(x-h)^2+k$, where:

• $a$: vertical stretch/compression; negative = reflection over x-axis
• $h$: horizontal shift ($h>0$ right, $h<0$ left)
• $k$: vertical shift ($k>0$ up, $k<0$ down)
• Vertex = $(h,k)$

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27. Analyze $f(x)=3(x+2)^2+1$

Step1: Identify $a,h,k$

$a=3$, $h=-2$, $k=1$

Step2: Describe transformations

Vertical stretch by 3, left 2, up 1

Step3: Find vertex

Vertex = $(-2,1)$

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28. Analyze $f(x)=-4(x+1)^2-5$

Step1: Identify $a,h,k$

$a=-4$, $h=-1$, $k=-5$

Step2: Describe transformations

Reflect over x-axis, vertical stretch by 4, left 1, down 5

Step3: Find vertex

Vertex = $(-1,-5)$

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29. Analyze $f(x)=-2x^2+5$

Step1: Identify $a,h,k$

$a=-2$, $h=0$, $k=5$

Step2: Describe transformations

Reflect over x-axis, vertical stretch by 2, up 5

Step3: Find vertex

Vertex = $(0,5)$

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30. Analyze $f(x)=\frac{1}{2}(x-1)^2$

Step1: Identify $a,h,k$

$a=\frac{1}{2}$, $h=1$, $k=0$

Step2: Describe transformations

Vertical compression by $\frac{1}{2}$, right 1

Step3: Find vertex

Vertex = $(1,0)$

Answer:

1. For $f(x)=3(x+2)^2+1$:

• Transformations: Vertically stretched by a factor of 3, shifted left 2 units and up 1 unit from $f(x)=x^2$.
• Vertex: $(-2, 1)$

1. For $f(x)=-4(x+1)^2-5$:

• Transformations: Reflected over the x-axis, vertically stretched by a factor of 4, shifted left 1 unit and down 5 units from $f(x)=x^2$.
• Vertex: $(-1, -5)$

1. For $f(x)=-2x^2+5$:

• Transformations: Reflected over the x-axis, vertically stretched by a factor of 2, shifted up 5 units from $f(x)=x^2$.
• Vertex: $(0, 5)$

1. For $f(x)=\frac{1}{2}(x-1)^2$:

• Transformations: Vertically compressed by a factor of $\frac{1}{2}$, shifted right 1 unit from $f(x)=x^2$.
• Vertex: $(1, 0)$