What are key features of the function f(x) = 5x² + 20x – 105? Select all the true statements.
A. The x-intercepts are –7 and 3.
B. The vertex is (0, –105).
C. The domain is –7 ≤ x ≤ 3.
D. The range is y ≥ –125.
E. The y-intercept is –125.
Key Features of the Quadratic Function f(x) = 5x² + 20x – 105
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To analyze the function f(x) = 5x² + 20x – 105, we can determine its key features.
1. Identify the vertex: The vertex form of a quadratic can be found using the formula ( x = -frac{b}{2a} ). Here, ( a = 5 ) and ( b = 20 ).
[
x = -frac{20}{2(5)} = -2
]
Now plug ( x = -2 ) back into the function to find the y-coordinate:
[
f(-2) = 5(-2)² + 20(-2) – 105 = 5(4) – 40 – 105 = 20 – 40 – 105 = -125
]
Thus, the vertex is ((-2, -125)).
2. Find the x-intercepts: To find x-intercepts, set ( f(x) = 0 ):
[
5x² + 20x – 105 = 0 quad Rightarrow quad x² + 4x – 21 = 0
]
Factor or use the quadratic formula to find the roots:
[
x = frac{-4 pm sqrt{(4)² – 4(1)(-21)}}{2(1)} =
To analyze the function ( f(x) = 5x^2 + 20x – 105 ), let’s evaluate each statement step by step.
1. Finding the vertex:
The vertex ( x ) coordinate is given by ( x = -frac{b}{2a} ) where ( a = 5 ) and ( b = 20 ).
[
x = -frac{20}{2 times 5} = -2
]
Now, substituting ( x = -2 ) back into the function to find the ( y ) coordinate:
[
f(-2) = 5(-2)^2 + 20(-2) – 105 = 5(4) – 40 – 105 = 20 – 40 – 105 = -125
]
So, the vertex is ((-2, -125)).
2. Finding the x-intercepts:
To find the x-intercepts, set ( f(x) = 0 ):
[
5x^2 + 20x – 105 = 0
]
This can be simplified to ( x^2 + 4x – 21 = 0) by dividing the entire equation by 5. Factoring gives:
[
(x + 7)(x – 3) = 0