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.