Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=4 x^{2}+8 x-3 $$

Short answer

The graph of the function \(y=4x^{2}+8x-3\) is a parabola opening upwards with the vertex at (-1,-7).

Step-by-step solution

Identify a, b and c

For the quadratic function \(y=ax^{2}+bx+c\), identify the values of a, b, and c. In the given equation \(y=4x^{2}+8x-3\), the values of a, b, and c are 4, 8, and -3 respectively.

Calculate the vertex

Calculate the x-coordinate of the vertex using the formula \(-\frac{b}{2a}\). Substituting the values of b and a gives \(-\frac{8}{2*4}\) which simplifies to -1. Then substitute -1 into the original equation to find the y-coordinate of the vertex. The vertex is (-1,-7).

Plot the vertex and sketch the graph

Plot the vertex (-1,-7) on a graph. As the coefficient of \(x^{2}\) is positive, the parabola opens upwards. Sketch the graph of the quadratic function keeping in mind the vertex and the shape of the parabola.