Question

Consider the function y equals 4 x squared plus 8 x minus 31.
​(a) Give the coordinates of the vertex of the graph of the function.
​(b) Graph the function on a window that includes the vertex.

Answer 1

To analyze the function \( y = 4x^2 + 8x - 31 \), we need to find the vertex and potentially plot the graph.

### (a) Finding the vertex

The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the vertex formula:

\[
x = -\frac{b}{2a}
\]

In your function, the coefficients are:
- \( a = 4 \)
- \( b = 8 \)
- \( c = -31 \)

Now, substituting these values into the vertex formula:

\[
x = -\frac{8}{2 \times 4} = -\frac{8}{8} = -1
\]

Now, to find the y-coordinate of the vertex, substitute \( x = -1 \) back into the original function.

\[
y = 4(-1)^2 + 8(-1) - 31
\]
\[
y = 4(1) - 8 - 31
\]
\[
y = 4 - 8 - 31
\]
\[
y = -35
\]

Therefore, the coordinates of the vertex are:

\[
\text{Vertex: } (-1, -35)
\]

### (b) Graphing the function

To graph the function \( y = 4x^2 + 8x - 31 \), it is helpful to understand the general shape of the graph:
- The parabola opens upwards (since \( a = 4 > 0 \)).
- The vertex is the lowest point, located at \( (-1, -35) \).

We would typically want to create a table of values around the vertex to plot other points. Here are a few values:

\[
\begin{align*}
x & : -3 & -2 & -1 & 0 & 1 \\
y & : 4(-3)^2 + 8(-3) - 31 & 4(-2)^2 + 8(-2) - 31 & 4(-1)^2 + 8(-1) - 31 & 4(0)^2 + 8(0) - 31 & 4(1)^2 + 8(1) - 31 \\
& = 36 - 24 - 31 & = 16 - 16 - 31 & = 4 - 8 - 31 & = 0 - 31 & = 4 + 8 - 31 \\
& = -19 & = -31 & = -35 & = -31 & = -19 \\
\end{align*}
\]

Plotting these points would result in the following coordinate pairs:
- \( (-3, -19) \)
- \( (-2, -31) \)
- \( (-1, -35) \) (the vertex)
- \( (0, -31) \)
- \( (1, -19) \)

You can plot these points on a graph, along with the axis, to show the parabola.

**Window Settings for the Graph:**
- X-axis range: From \( -4 \) to \( 2 \) (to include vertex and points around it)
- Y-axis range: From \( -40 \) to \( 0 \) (to capture the vertex low point and other relevant y-values)

The resulting parabola will look something like this:
- There will be symmetry about the line \( x = -1 \) (the x-coordinate of the vertex).
- The lowest point is the vertex located at (-1, -35).

### Summary:
- **Vertex Coordinates:** (-1, -35)
- **Graphing:** Use a quadratic graphing tool or software with points above to visualize the graph.