The graph of the function f(x) = (x – 4)(x + 1) is shown below.
On a coordinate plane, a parabola opens up. It goes through (negative 1, 0), has a vertex at (1.75, negative 6.2), and goes through (4, 0).
Which statement about the function is true?
The function is increasing for all real values of x where
x 4.
The function is decreasing for all real values of x where
–1 < x < 4.
The function is decreasing for all real values of x where
x < 1.5.
Answers
Given : f(x) = (x – 4)(x + 1)
To Find : Which statement about the function is true?
It goes through (negative 1, 0)
has a vertex at (1.75, negative 6.2)
goes through (4, 0).
The function is increasing for all real values of x where x > 4.
The function is decreasing for all real values of x where –1 < x < 4.
The function is decreasing for all real values of x where x < 1.5.
Solution:
f(x) = (x – 4)(x + 1)
x = - 1 => f(x) = 0 hence goes through (negative 1, 0)
x = 4 => f(x) = 0 hence goes through (4, 0)
=> f(x) = x² - 3x - 4
= (x - 1.5)² - 6.25
Hence vertex is ( 1.5 , 6.25)
has a vertex at (1.75, negative 6.2) - FALSE
f'(x) = 2x - 3
The function is increasing if 2x - 3 > 0
=> x > 3/2
The function is increasing for all real values of x where x > 1.5
Hence this is also true The function is increasing for all real values of x where x > 4
The function is decreasing for all real values of x where x < 1.5
The function is decreasing for all real values of x where x < 1.5. TRUE
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