Question

if (a > 0), which of the following could be an equation of the graph shown in the figure to the right?

(a) (y = ax^2)
(b) (y = x^2 - a)
(c) (y = x^2 + a)
(d) (y = (x - a)^2)
(e) (y = (x + a)^2)

Explanation:

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<reasoning>

Analyze the graph features

\[

$$\begin{aligned} &\text{Vertex is on the } y\text{-axis above the origin: } (0, k) \text{ where } k > 0\\ &\text{Parabola opens upward: } y = x^2 + k \end{aligned}$$

\]

Evaluate the options given \(a > 0\)

\[

$$\begin{aligned} &\text{(A) } y = ax^2 \implies \text{Vertex is at } (0,0)\\ &\text{(B) } y = x^2 - a \implies \text{Vertex is at } (0, -a) \text{ with } -a < 0\\ &\text{(C) } y = x^2 + a \implies \text{Vertex is at } (0, a) \text{ with } a > 0\\ &\text{(D) } y = (x - a)^2 \implies \text{Vertex is at } (a, 0) \text{ on the positive } x\text{-axis}\\ &\text{(E) } y = (x + a)^2 \implies \text{Vertex is at } (-a, 0) \text{ on the negative } x\text{-axis} \end{aligned}$$

\]

Match graph with correct option

\[
\text{The graph vertex is at } (0, k) \text{ with } k > 0 \implies y = x^2 + a
\]
</reasoning>

<answer>
<mcq-option>(A) \(y = ax^2\)</mcq-option>
<mcq-option>(B) \(y = x^2 - a\)</mcq-option>
<mcq-correct>(C) \(y = x^2 + a\)</mcq-correct>
<mcq-option>(D) \(y = (x - a)^2\)</mcq-option>
<mcq-option>(E) \(y = (x + a)^2\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Analyze the graph features

\[

$$\begin{aligned} &\text{Vertex is on the } y\text{-axis above the origin: } (0, k) \text{ where } k > 0\\ &\text{Parabola opens upward: } y = x^2 + k \end{aligned}$$

\]

Evaluate the options given \(a > 0\)

\[

$$\begin{aligned} &\text{(A) } y = ax^2 \implies \text{Vertex is at } (0,0)\\ &\text{(B) } y = x^2 - a \implies \text{Vertex is at } (0, -a) \text{ with } -a < 0\\ &\text{(C) } y = x^2 + a \implies \text{Vertex is at } (0, a) \text{ with } a > 0\\ &\text{(D) } y = (x - a)^2 \implies \text{Vertex is at } (a, 0) \text{ on the positive } x\text{-axis}\\ &\text{(E) } y = (x + a)^2 \implies \text{Vertex is at } (-a, 0) \text{ on the negative } x\text{-axis} \end{aligned}$$

\]

Match graph with correct option

\[
\text{The graph vertex is at } (0, k) \text{ with } k > 0 \implies y = x^2 + a
\]
</reasoning>

<answer>
<mcq-option>(A) \(y = ax^2\)</mcq-option>
<mcq-option>(B) \(y = x^2 - a\)</mcq-option>
<mcq-correct>(C) \(y = x^2 + a\)</mcq-correct>
<mcq-option>(D) \(y = (x - a)^2\)</mcq-option>
<mcq-option>(E) \(y = (x + a)^2\)</mcq-option>
</answer>

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