Answer:
see below
Step-by-step explanation:
f(x) = 5x^2 + 2x − 3
Factor
f(x) = (x + 1) (5 x - 3)
To find the x intercepts, set f(x) =0
0 = (x+1) ( 5x-3)
Using the zero product property
x+1 = 0 5x-3 =0
x=-1 5x=3
x = 3/5
The x intercepts are (-1,0) and (3/5,0)
When x goes to -∞ the 5x^2 term dominates
f( -∞) = 5 ( -∞) ^2 = 5 ( ∞) = ∞
As x goes to -∞ f(x) goes to ∞
When x goes to ∞ the 5x^2 term dominates
f( ∞) = 5 ( ∞) ^2 = 5 ( ∞) = ∞
As x goes to ∞ f(x) goes to ∞
We need the end behavior, the x intercepts and the vertex to graph
We know this opens upward since the coefficient of x^2 is positive
The vertex is 1/2 way between the zeros
(-1+3/5) /2 = (-2/5) /2 = -1/5
Substitute this into f(x) to find the minimum
f(-1/5) = (-1/5 + 1) (5 *-1/5 - 3) = 4/5 * (-4) = -16/5