Math Problem Statement
The table gives selected values of the function f(x). Part A: (i) Determine the rate of change for function f on the interval [3, 5]. (ii) How is the rate of change in part (i) displayed on the graph of a function? Part B: (i) Determine the rate of change for function f on the interval [3, 6]. (ii) How is the rate of change in part (i) displayed on the graph of a function? Part C: Compare the average rates of change for f on the intervals [3, 5] and [3, 6].
Solution
Let's break down the questions from the table and calculate the answers step by step.
The table provides selected values of the function :
x & 3 & 4 & 5 & 6 & 7 \\ \hline f(x) & 0 & -3 & -4 & -3 & 0 \\ \end{array}$$ ### Part A: (i) **Determine the rate of change for function $$f$$ on the interval [3, 5]:** The rate of change is calculated using the formula for the slope between two points: $$\text{Rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$$ For the interval [3, 5]: - $$x_1 = 3$$, $$f(3) = 0$$ - $$x_2 = 5$$, $$f(5) = -4$$ $$\text{Rate of change} = \frac{-4 - 0}{5 - 3} = \frac{-4}{2} = -2$$ So, the rate of change is **-2**. (ii) **How is the rate of change in part (i) displayed on the graph of the function?** Since the rate of change is negative, the function is **decreasing** on the interval [3, 5], meaning that the graph would slope downward. ### Part B: (i) **Determine the rate of change for function $$f$$ on the interval [3, 6]:** Using the same formula for the rate of change: For the interval [3, 6]: - $$x_1 = 3$$, $$f(3) = 0$$ - $$x_2 = 6$$, $$f(6) = -3$$ $$\text{Rate of change} = \frac{-3 - 0}{6 - 3} = \frac{-3}{3} = -1$$ So, the rate of change is **-1**. (ii) **How is the rate of change in part (i) displayed on the graph of a function?** Since the rate of change is negative but less steep than in part A, the function is still **decreasing**, but more gradually than in the interval [3, 5]. ### Part C: **Compare the average rates of change for $$f$$ on the intervals [3, 5] and [3, 6]:** - The rate of change on [3, 5] is **-2**. - The rate of change on [3, 6] is **-1**. The interval [3, 5] has a **steAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Rate of Change
Slope of a Line
Functions
Formulas
Rate of Change = (f(x2) - f(x1)) / (x2 - x1)
Theorems
Slope Formula
Suitable Grade Level
Grades 8-10