To analyze the quadratic function \( f(x) = 6x - x^2 + 16 \), we can first rewrite it in standard form. Rearranging the terms gives us:
\[ f(x) = -x^2 + 6x + 16 \]
Step 1: Find the Vertex
The vertex of a quadratic function in the form \( ax^2 + bx + c \) can be found using the vertex formula:
\[ x = -\frac{b}{2a} \]
Here, \( a = -1 \) and \( b = 6 \). Substituting these values into the formula gives:
\[ x = -\frac{6}{2 \times -1} = 3 \]
To find the \( y \)-coordinate of the vertex, plug \( x = 3 \) back into the function:
\[ f(3) = -3^2 + 6(3) + 16 = -9 + 18 + 16 = 25 \]
Thus, the vertex is \( (3, 25) \).
Step 2: Find the x-Intercepts
The x-intercepts occur when \( f(x) = 0 \):
\[ 0 = -x^2 + 6x + 16 \]
Rearranging gives:
\[ x^2 - 6x - 16 = 0 \]
Now we use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1 \), \( b = -6 \), and \( c = -16 \):
\[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot -16}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{6 \pm \sqrt{36 + 64}}{2} = \frac{6 \pm \sqrt{100}}{2} = \frac{6 \pm 10}{2} \]
This gives us:
\[ x = \frac{16}{2} = 8 \quad \text{and} \quad x = \frac{-4}{2} = -2 \]
So the x-intercepts are \( (8, 0) \) and \( (-2, 0) \).
Step 3: Find the y-Intercept
The y-intercept occurs when \( x = 0 \):
\[ f(0) = -0^2 + 6(0) + 16 = 16 \]
So the y-intercept is \( (0, 16) \).
Step 4: Axis of Symmetry
The axis of symmetry for the parabola is given by:
\[ x = 3 \]
Step 5: Domain and Range
For a quadratic function, the domain is all real numbers, which in interval notation is:
\[ \text{Domain: } (-\infty, \infty) \]
Now to find the range, since the parabola opens downwards (as \( a = -1 < 0 \)), the maximum value is at the vertex \( y = 25 \) and goes down to negative infinity. Thus, the range is:
\[ \text{Range: } (-\infty, 25] \]
Summary
- The equation of the parabola's axis of symmetry is: \( x = 3 \)
- The domain of the function is: \( (-\infty, \infty) \)
- The range of the function is: \( (-\infty, 25] \)
You can plot the vertex (3, 25), the x-intercepts (8, 0) and (-2, 0), and the y-intercept (0, 16) to graph the parabola, ensuring it opens downwards. If you need a graphing tool to visualize this, you can use online graphing tools like Desmos or GeoGebra.