Question

Use the vertex and intercepts to sketch the graph of the
quadratic function. Give the equation of the parabola's
axis of symmetry. Use the graph to determine
the function's domain and range.

f(x) = x^2-8x-9

Use the graphing tool to graph the equation. Use the
vertex and one of the intercepts when drawing the
graph.


The axis of symmetry is blank
(Type an equation.)

The domain of f is blank
(Type your answer in interval notation.)

The range of f is blank
(Type your answer in interval notation.)

Answer 1

To analyze the quadratic function \( f(x) = x^2 - 8x - 9 \), we will find its vertex, intercepts, and then use this information to sketch the graph.

### Step 1: Find the Vertex

The vertex form of a quadratic function is given by:

\[
f(x) = a(x - h)^2 + k
\]

To find the vertex using the standard form \( f(x) = ax^2 + bx + c \), we can use the formula for the x-coordinate of the vertex:

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

For our function:
- \( a = 1 \)
- \( b = -8 \)
- \( c = -9 \)

Substituting in the values:

\[
h = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]

Now substitute \( x = 4 \) back into the function to find the y-coordinate \( k \):

\[
k = f(4) = 4^2 - 8 \cdot 4 - 9 = 16 - 32 - 9 = -25
\]

So the vertex is at:

\[
(4, -25)
\]

### Step 2: Find the x-intercepts

To find the x-intercepts, we set \( f(x) = 0 \):

\[
x^2 - 8x - 9 = 0
\]

We can solve this using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1}
\]

Calculating the discriminant:

\[
b^2 - 4ac = 64 + 36 = 100
\]

Now substituting back:

\[
x = \frac{8 \pm \sqrt{100}}{2} = \frac{8 \pm 10}{2}
\]

This gives us the two solutions:

\[
x = \frac{18}{2} = 9 \quad \text{and} \quad x = \frac{-2}{2} = -1
\]

The x-intercepts are \( (9, 0) \) and \( (-1, 0) \).

### Step 3: Find the y-intercept

To find the y-intercept, we set \( x = 0 \):

\[
f(0) = 0^2 - 8 \cdot 0 - 9 = -9
\]

The y-intercept is \( (0, -9) \).

### Step 4: Sketch the Graph

Now we have:

- Vertex: \( (4, -25) \)
- X-intercepts: \( (9, 0) \) and \( (-1, 0) \)
- Y-intercept: \( (0, -9) \)

Using this information, we can sketch the parabola, which opens upwards.

### Step 5: Axis of Symmetry

The axis of symmetry of a parabola given by the equation \( ax^2 + bx + c \) is given by the line:

\[
x = h
\]

Thus, the equation of the axis of symmetry is:

\[
\text{Axis of symmetry: } x = 4
\]

### Step 6: Domain and Range

The domain of any quadratic function is all real numbers:

\[
\text{Domain of } f: (-\infty, \infty)
\]

The range depends on the vertex. Since the vertex is the minimum point and the parabola opens upwards, the range starts from the y-coordinate of the vertex:

\[
\text{Range of } f: [-25, \infty)
\]

### Final Answers

- The axis of symmetry is \( x = 4 \).
- The domain of \( f \) is \( (-\infty, \infty) \).
- The range of \( f \) is \( [-25, \infty) \).