Given
The function is given as
[tex]f(x)=5x^2+2x-3[/tex]Explanation
a. To determine the x-intercepts of the function
Substitute the y -coordinate equal to 0.
[tex]5x^2+2x-3=0[/tex]
Factorize the equation.
[tex]\begin{gathered} (x-\frac{3}{5})(x+1)=0 \\ x=\frac{3}{5},-1. \end{gathered}[/tex]
AnswerThe x-intercepts are
[tex]x=\frac{3}{5},-1[/tex]
b. The vertex of the function is determined as
[tex]x=-\frac{b}{2a}[/tex]
Substitute the values.
[tex]x=-\frac{2}{5\times2}=-0.2[/tex]
Now substitute the value of x in the function.
[tex]\begin{gathered} 5(-0.2)^2+2(-0.2)-3=f(-0.2) \\ 5\times0.04-0.4-3=f(-0.2) \\ 0.2-0.4-3=-3.2 \end{gathered}[/tex]Answer
The coordinates of vertex is
[tex](-0.2,-3.2)[/tex]
The condition for vertex to be maximum or minimum,
If a>0, the parabola opens upward and the vertex is a minimum.
If a<0, the parabola opens downward, and the vertex is a maximum.
As a for he given function is 5 which is positive.
Then the vertex is a minimum.
c. The steps for graph the function.
We know the x-intercepts of the function and substitute in the graph.
Also plot the points of the vertex in the graph.
Answer
Then the graph is determined as
