Examine the following graph of the linear function f(x)=2x+1 and the quadratic function g(x)=−116(x+3)2+6.
Graph of g of x is a downward opening parabola that passes through points (negative 4, 5), (negative 3, 6), & (1, 5). f of x is a line that passes through points (negative 4, negative 7), (0, 1), & (3, 7).
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Which statements about the functions are true?
There is more than one correct answer. Select all correct answers.
Over the interval 4≤x≤6, the average rate of change of g(x) is faster than the average rate of change of f(x).
Over the interval −8≤x≤−7, the average rate of change of g(x) is slower than the average rate of change of f(x).
Over the interval 0≤x≤3, the average rate of change of g(x) is the same as the average rate of change of f(x).
Over the interval \(-8 \le x \le -3\textsf{,}\) the average rate of change of \(g(x)\) is slower than the average rate of change of \(f(x)\textsf{.}\)
Over the interval \(4 \le x \le 6\textsf{,}\) the average rate of change of \(g(x)\) is slower than the average rate of change of \(f(x)\textsf{.}\)
Over the interval \(-3 \le x \le 2\textsf{,}\) the average rate of change of \(g(x)\) is faster than the average rate of change of \(f(x)\textsf{.}\)
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To compare the average rate of change of the two functions, we can find the slopes of their lines.
The linear function f(x) = 2x + 1 has a slope of 2.
The quadratic function g(x) = -1/16(x + 3)^2 + 6 can be rewritten as g(x) = -1/16(x^2 + 6x + 9) + 6.
Expanding the equation, we get g(x) = -1/16x^2 - 3/8x - 43/16.
Comparing the coefficients, we see that the coefficient of x^2 is -1/16, so the slope of the quadratic function is -1/8.
Now, let's analyze each statement:
1. Over the interval 4 ≤ x ≤ 6, the average rate of change of g(x) is faster than the average rate of change of f(x).
To compare the slopes, we see that the slope of g(x) (-1/8) is smaller than the slope of f(x) (2). Therefore, this statement is false.
2. Over the interval -8 ≤ x ≤ -7, the average rate of change of g(x) is slower than the average rate of change of f(x).
Since the interval is not within the given points for f(x) and g(x), we cannot make a conclusion about the average rate of change. Therefore, this statement is not applicable.
3. Over the interval 0 ≤ x ≤ 3, the average rate of change of g(x) is the same as the average rate of change of f(x).
The interval 0 ≤ x ≤ 3 is within the given points for both f(x) and g(x). Since the slopes of f(x) and g(x) are different (2 and -1/8, respectively), this statement is false.
4. Over the interval -8 ≤ x ≤ -3, the average rate of change of g(x) is slower than the average rate of change of f(x).
The interval -8 ≤ x ≤ -3 is within the given points for both f(x) and g(x).
Since the slopes of f(x) and g(x) are 2 and -1/8, respectively, and 2 > -1/8, the average rate of change of f(x) is faster than that of g(x). Therefore, this statement is true.
5. Over the interval 4 ≤ x ≤ 6, the average rate of change of g(x) is slower than the average rate of change of f(x).
Since the interval is not within the given points for f(x) and g(x), we cannot make a conclusion about the average rate of change. Therefore, this statement is not applicable.
6. Over the interval -3 ≤ x ≤ 2, the average rate of change of g(x) is faster than the average rate of change of f(x).
The interval -3 ≤ x ≤ 2 is within the given points for both f(x) and g(x). Since the slopes of f(x) and g(x) are 2 and -1/8, respectively, and 2 > -1/8, the average rate of change of f(x) is faster than that of g(x). Therefore, this statement is false.
In conclusion, the true statements are:
- Over the interval -8 ≤ x ≤ -3, the average rate of change of g(x) is slower than the average rate of change of f(x).
- Over the interval -3 ≤ x ≤ 2, the average rate of change of g(x) is faster than the average rate of change of f(x).