The function provided is $f(x) = -x^2 + 8x - 9$, which is a quadratic equation. Let's graph it step by step:

Step 1: Identify the Vertex

The general quadratic form is $f(x) = ax^2 + bx + c$. Here, $a = -1$, $b = 8$, and $c = -9$.

The vertex of a parabola is given by: $x_{\text{vertex}} = -\frac{b}{2a}$

Substitute $a = -1$ and $b = 8$: $x_{\text{vertex}} = -\frac{8}{2(-1)} = \frac{8}{2} = 4$

To find the $y$-coordinate of the vertex, substitute $x = 4$ into the equation: $f(4) = -(4)^2 + 8(4) - 9$ $f(4) = -16 + 32 - 9 = 7$

So, the vertex is $(4, 7)$.

Step 2: Plot Another Point

Choose another $x$-value close to the vertex, such as $x = 5$.

Substitute $x = 5$ into the equation: $f(5) = -(5)^2 + 8(5) - 9$ $f(5) = -25 + 40 - 9 = 6$

So, another point is $(5, 6)$.

You can also reflect this point over the axis of symmetry (which is $x = 4$) to find $(3, 6)$.

Step 3: Plot the Parabola

• Plot the vertex $(4, 7)$.
• Plot the point $(5, 6)$ and its reflection $(3, 6)$.
• Continue finding additional points if needed (e.g., $x = 6$ or $x = 2$) and plot them to complete the parabola.

The graph opens downward because $a = -1$ is negative.

Would you like additional points calculated or any further details? 😊

5 Related Questions:

1. How do you find the axis of symmetry for a parabola?
2. What does the sign of $a$ tell us about the direction of the parabola?
3. How can you find the x-intercepts of this parabola?
4. What is the significance of the vertex in the graph of a parabola?
5. How do reflections across the axis of symmetry work?

Tip:

To quickly check the accuracy of your graph, plug in additional $x$-values into the function to verify the points!