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 \( [-25, \infty) \).
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