the graph represents the function (f(x) = 10(2)^x).

how would the graph change if the (b) value in the equation is decreased but remains greater than 1?
check all that apply.

the graph will begin at a lower point on the (y)-axis.
the graph will increase at a faster rate.
the graph will increase at a slower rate.
the (y)-values will continue to increase as (x)-increases.
the (y)-values will each be less than their corresponding (x)-values.

Question

the graph represents the function (f(x) = 10(2)^x).

how would the graph change if the (b) value in the equation is decreased but remains greater than 1?
check all that apply.

the graph will begin at a lower point on the (y)-axis.
  the graph will increase at a faster rate.
  the graph will increase at a slower rate.
  the (y)-values will continue to increase as (x)-increases.
  the (y)-values will each be less than their corresponding (x)-values.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponential Functions", "y-intercept", "Exponential Growth", "Common Ratio" ], "new_concepts": [], "current_concepts": [ "Exponential Functions", "y-intercept", "Exponential Growth", "Common Ratio" ] } ## Analyze the effect on the y-intercept $$ \begin{aligned} &f(x) = a \cdot b^x \implies f(0) = a \cdot b^0 = a\ &f(0) = 10 \cdot (2)^0 = 10\ &\text{Since } a = 10 \text{ remains unchanged, the } y\text{-intercept remains at } (0, 10). \end{aligned} $$ ## Analyze the effect on the growth rate $$ \begin{aligned} &1 < b_{\text{new}} < 2\ &\text{Since } b_{\text{new}} > 1, \text{ the function continues to increase as } x \text{ increases.}\ &\text{Since } b_{\text{new}} < 2, \text{ the base is smaller, so the graph increases at a slower rate.} \end{aligned} $$ ## Evaluate the given statements $$ \begin{aligned} &\text{1. "The graph will begin at a lower point on the y-axis" } \implies \text{False (y-intercept is unchanged).}\ &\text{2. "The graph will increase at a faster rate" } \implies \text{False (base decreased).}\ &\text{3. "The graph will increase at a slower rate" } \implies \text{True (base decreased but } > 1\text{).}\ &\text{4. "The y-values will continue to increase as x-increases" } \implies \text{True (base remains } > 1\text{).}\ &\text{5. "The y-values will each be less than their corresponding x-values" } \implies \text{False (e.g., at } x=0, y=10 > 0\text{).} \end{aligned} $$

The graph will begin at a lower point on the y-axis.

The graph will increase at a faster rate.

The graph will increase at a slower rate.

The y-values will continue to increase as x-increases.

The y-values will each be less than their corresponding x-values.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the effect on the y-intercept

Analyze the effect on the growth rate

Evaluate the given statements

Answer: