the table represents $f(x)$.

what type of function could be used to represent $f(x)$?

- exponential
- linear
- quadratic
- square root

Question

the table represents $f(x)$.

what type of function could be used to represent $f(x)$?

- exponential
- linear
- quadratic
- square root

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Finite Differences", "Function Modeling" ], "new_concepts": [ "Exponential Functions", "Common Ratio" ], "current_concepts": [ "Finite Differences", "Function Modeling", "Exponential Functions", "Common Ratio" ] } ## Analyze the rate of change Using the *Finite Differences* and *Function Modeling* knowledge points $$ \begin{aligned} &\Delta x = 1 - 0 = 1\ &\text{First differences of } f(x):\ &f(0) - f(-1) = -4 - (-2) = -2\ &f(1) - f(0) = -8 - (-4) = -4\ &f(2) - f(1) = -16 - (-8) = -8 \end{aligned} $$ ## Determine the ratio of consecutive terms Using the *Function Modeling* knowledge point $$ \begin{aligned} &\frac{f(0)}{f(-1)} = \frac{-4}{-2} = 2\ &\frac{f(1)}{f(0)} = \frac{-8}{-4} = 2\ &\frac{f(2)}{f(1)} = \frac{-16}{-8} = 2 \end{aligned} $$ ## Identify the function type Since there is a constant ratio of $2$ between consecutive terms as $x$ increases by $1$, the function grows by a constant factor. This behavior characterizes an exponential function of the form $f(x) = a \cdot b^x$. Here, $f(x) = -4 \cdot 2^x$.

Exponential

Linear

Quadratic

Square root

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

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QUESTION IMAGE

Question

Explanation:

Analyze the rate of change

Determine the ratio of consecutive terms

Identify the function type

Answer: