Question

The graph of y = 3x2 + 4x − 7 is shown below. What are the zeros of the function (as exact values), the y-intercept, and the maximum or minimum value of the function (rounded to the nearest tenth)?Hint: Use a graphing utility or rewrite the function in different forms to help you answer these questions.

Answer 1

Zeros: -7/3, 1; y-intercept: (0,-7) ;minimum: -8.3

1) Considering the quadratic function y=3x²+4x-7, we can find their zeros by solving it:

`\begin{gathered} y=3x^2+4x-7 \\ x=\frac{-b\pm\sqrt[]{\Delta}}{2a}=\frac{-4\pm\sqrt[]{(16)-4(3)(-7)}}{2(3)}= \\ x_1=1 \\ x_2=-(7)/(3) \end{gathered}`

Note that the zeros can also be called "roots" and since the parabola hits the x-axis twice we have two Real roots.

2) Let's continue. Looking at the function we can state that the y-intercept is given by the number -7, or the coefficient "c". As a point, we have (0,-7) as the y-intercept

2.2) We can also find the minimum by using this formula since a >0:

`h=(-b)/(2a)=(-4)/(2(3))=-(4)/(6)=-(2)/(3)\approx-0.7`

This is the x-coordinate of the minimum. We need to plug into the function to get the y-coordinate:

`k=3(-(2)/(3))^2+4(-(2)/(3))-7\approx-8.3`

So the minimum point is located at (-0.7, -8.3)

3) Hence, the answer is:

Zeros: -7/3, 1; y-intercept: (0,-7) ;minimum: -8.3