Question

Graph the function f(x)=5x2–6.

Answer 1

To graph the function \( f(x) = 5x^2 - 6 \), follow these steps:

1. **Identify the Type of Function**: This is a quadratic function in the form \( f(x) = ax^2 + bx + c \) where \( a = 5 \), \( b = 0 \), and \( c = -6 \). Since \( a > 0 \), the parabola opens upwards.

2. **Find the Vertex**: The vertex of a quadratic function in standard form can be found using the formula
\[
x = -\frac{b}{2a}.
\]
In this case, since \( b = 0 \), the vertex is at
\[
x = 0.
\]
To find the y-coordinate of the vertex, substitute \( x = 0 \) into the function:
\[
f(0) = 5(0)^2 - 6 = -6.
\]
So the vertex is at the point \( (0, -6) \).

3. **Determine the Axis of Symmetry**: The axis of symmetry for the parabola is the vertical line that passes through the vertex, which is \( x = 0 \).

4. **Find Additional Points**: To get a better representation of the parabola, calculate a few more points by selecting some x-values. Here are some calculations:

- For \( x = -2 \):
\[
f(-2) = 5(-2)^2 - 6 = 5(4) - 6 = 20 - 6 = 14 \quad \Rightarrow (-2, 14)
\]
- For \( x = -1 \):
\[
f(-1) = 5(-1)^2 - 6 = 5(1) - 6 = 5 - 6 = -1 \quad \Rightarrow (-1, -1)
\]
- For \( x = 1 \):
\[
f(1) = 5(1)^2 - 6 = 5(1) - 6 = 5 - 6 = -1 \quad \Rightarrow (1, -1)
\]
- For \( x = 2 \):
\[
f(2) = 5(2)^2 - 6 = 5(4) - 6 = 20 - 6 = 14 \quad \Rightarrow (2, 14)
\]

5. **Plot the Points and Parabola**: You now have several points:
- Vertex: \( (0, -6) \)
- Points: \( (-2, 14) \), \( (-1, -1) \), \( (1, -1) \), \( (2, 14) \)

6. **Draw the Graph**: Plot the vertex and the other points on a graph. Draw a smooth curve through these points, ensuring it opens upwards.

### Summary of Key Points
- Vertex: \( (0, -6) \)
- Axis of symmetry: \( x = 0 \)
- Points: \( (-2, 14) \), \( (-1, -1) \), \( (1, -1) \), \( (2, 14) \)

By plotting these points and drawing the curve, you will have a complete graph of the function \( f(x) = 5x^2 - 6 \).