Question

which functions represent exponential growth?
\\(y = f(x)\\)
\\(y = h(x)\\)
\\(y = g(x)\\)
\\(y = k(x)\\)

Explanation:

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<reasoning>

Identify exponential growth behavior

Using the Exponential Growth and Decay Behavior knowledge point
\[

$$\begin{aligned} &\text{Exponential growth functions of the form } y = ab^x \text{ (with } a > 0, b > 1\text{) increase as } x \to \infty.\\ &\text{Graphically, as we move from left to right, the curve rises and } y \to \infty. \end{aligned}$$

\]

Analyze each function graph

Using the Exponential Growth and Decay Behavior knowledge point
\[

$$\begin{aligned} &y = f(x) \text{ (orange curve): decreases as } x \text{ increases } \implies \text{Exponential Decay}\\ &y = g(x) \text{ (blue curve): decreases as } x \text{ increases } \implies \text{Exponential Decay}\\ &y = h(x) \text{ (green curve): increases as } x \text{ increases } \implies \text{Exponential Growth}\\ &y = k(x) \text{ (purple curve): increases as } x \text{ increases } \implies \text{Exponential Growth} \end{aligned}$$

\]

Select the correct options

Using the Exponential Growth and Decay Behavior knowledge point
\[

$$\begin{aligned} &\text{The functions representing exponential growth are } y = h(x) \text{ and } y = k(x). \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(y = f(x)\)</mcq-option>
<mcq-correct>\(y = h(x)\)</mcq-correct>
<mcq-option>\(y = g(x)\)</mcq-option>
<mcq-correct>\(y = k(x)\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify exponential growth behavior

Using the Exponential Growth and Decay Behavior knowledge point
\[

$$\begin{aligned} &\text{Exponential growth functions of the form } y = ab^x \text{ (with } a > 0, b > 1\text{) increase as } x \to \infty.\\ &\text{Graphically, as we move from left to right, the curve rises and } y \to \infty. \end{aligned}$$

\]

Analyze each function graph

Using the Exponential Growth and Decay Behavior knowledge point
\[

$$\begin{aligned} &y = f(x) \text{ (orange curve): decreases as } x \text{ increases } \implies \text{Exponential Decay}\\ &y = g(x) \text{ (blue curve): decreases as } x \text{ increases } \implies \text{Exponential Decay}\\ &y = h(x) \text{ (green curve): increases as } x \text{ increases } \implies \text{Exponential Growth}\\ &y = k(x) \text{ (purple curve): increases as } x \text{ increases } \implies \text{Exponential Growth} \end{aligned}$$

\]

Select the correct options

Using the Exponential Growth and Decay Behavior knowledge point
\[

$$\begin{aligned} &\text{The functions representing exponential growth are } y = h(x) \text{ and } y = k(x). \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(y = f(x)\)</mcq-option>
<mcq-correct>\(y = h(x)\)</mcq-correct>
<mcq-option>\(y = g(x)\)</mcq-option>
<mcq-correct>\(y = k(x)\)</mcq-correct>
</answer>

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