Question

which function has a vertex on the \\(y\\)-axis?

\\(f(x) = (x - 2)^2\\)
\\(f(x) = x(x + 2)\\)
\\(f(x) = (x - 2)(x + 2)\\)
\\(f(x) = (x + 1)(x - 2)\\)

Explanation:

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<reasoning>

Define condition for vertex on y-axis

Using the Quadratic Vertex knowledge point
\[
x_v = 0
\]

Analyze the first function

Using the Quadratic Vertex knowledge point
\[

$$\begin{aligned} &f(x) = (x-2)^2\\ &x_v = 2 e 0 \end{aligned}$$

\]

Analyze the second function

Using the Intercept Form and Quadratic Vertex knowledge points
\[

$$\begin{aligned} &f(x) = x(x+2)\\ &x_1 = 0,\quad x_2 = -2\\ &x_v = \frac{0 + (-2)}{2} = -1 e 0 \end{aligned}$$

\]

Analyze the third function

Using the Intercept Form and Quadratic Vertex knowledge points
\[

$$\begin{aligned} &f(x) = (x-2)(x+2)\\ &x_1 = 2,\quad x_2 = -2\\ &x_v = \frac{2 + (-2)}{2} = 0 \end{aligned}$$

\]

Analyze the fourth function

Using the Intercept Form and Quadratic Vertex knowledge points
\[

$$\begin{aligned} &f(x) = (x+1)(x-2)\\ &x_1 = -1,\quad x_2 = 2\\ &x_v = \frac{-1 + 2}{2} = 0.5 e 0 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(f(x) = (x - 2)^2\)</mcq-option>
<mcq-option>(B) \(f(x) = x(x + 2)\)</mcq-option>
<mcq-correct>(C) \(f(x) = (x - 2)(x + 2)\)</mcq-correct>
<mcq-option>(D) \(f(x) = (x + 1)(x - 2)\)</mcq-option>
</answer>

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"Quadratic Vertex"
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</post_analysis>

Answer:

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<reasoning>

Define condition for vertex on y-axis

Using the Quadratic Vertex knowledge point
\[
x_v = 0
\]

Analyze the first function

Using the Quadratic Vertex knowledge point
\[

$$\begin{aligned} &f(x) = (x-2)^2\\ &x_v = 2 e 0 \end{aligned}$$

\]

Analyze the second function

Using the Intercept Form and Quadratic Vertex knowledge points
\[

$$\begin{aligned} &f(x) = x(x+2)\\ &x_1 = 0,\quad x_2 = -2\\ &x_v = \frac{0 + (-2)}{2} = -1 e 0 \end{aligned}$$

\]

Analyze the third function

Using the Intercept Form and Quadratic Vertex knowledge points
\[

$$\begin{aligned} &f(x) = (x-2)(x+2)\\ &x_1 = 2,\quad x_2 = -2\\ &x_v = \frac{2 + (-2)}{2} = 0 \end{aligned}$$

\]

Analyze the fourth function

Using the Intercept Form and Quadratic Vertex knowledge points
\[

$$\begin{aligned} &f(x) = (x+1)(x-2)\\ &x_1 = -1,\quad x_2 = 2\\ &x_v = \frac{-1 + 2}{2} = 0.5 e 0 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(f(x) = (x - 2)^2\)</mcq-option>
<mcq-option>(B) \(f(x) = x(x + 2)\)</mcq-option>
<mcq-correct>(C) \(f(x) = (x - 2)(x + 2)\)</mcq-correct>
<mcq-option>(D) \(f(x) = (x + 1)(x - 2)\)</mcq-option>
</answer>

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