Questions: Consider the Quadratic function f(x)=x^2-6x-16. Its vertex is ( , ). Its largest x-intercept = . Enter the x-coordinate only. Its y-intercept = . Enter the y-coordinate only.

Solution Steps

To solve the given quadratic function \( f(x) = x^2 - 6x - 16 \):

1. Vertex: The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Once we have the x-coordinate, we can substitute it back into the function to get the y-coordinate.
2. Largest x-intercept: The x-intercepts (roots) of the quadratic function can be found using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The largest x-intercept is the larger of the two roots.
3. y-intercept: The y-intercept of the function is the value of the function when \( x = 0 \).

The vertex of the quadratic function \( f(x) = x^2 - 6x - 16 \) is calculated using the formula for the x-coordinate of the vertex:

\[ x = -\frac{b}{2a} = -\frac{-6}{2 \cdot 1} = 3.0 \]

Substituting \( x = 3.0 \) back into the function to find the y-coordinate:

\[ f(3) = 3^2 - 6 \cdot 3 - 16 = 9 - 18 - 16 = -25.0 \]

Thus, the vertex is \( (3.0, -25.0) \).

The x-intercepts are found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Calculating the discriminant:

\[ b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot (-16) = 36 + 64 = 100 \]

Now, applying the quadratic formula:

\[ x_1 = \frac{-(-6) + \sqrt{100}}{2 \cdot 1} = \frac{6 + 10}{2} = 8.0 \] \[ x_2 = \frac{-(-6) - \sqrt{100}}{2 \cdot 1} = \frac{6 - 10}{2} = -2.0 \]

The largest x-intercept is \( 8.0 \).

The y-intercept is found by evaluating the function at \( x = 0 \):

\[ f(0) = 0^2 - 6 \cdot 0 - 16 = -16 \]

Thus, the y-intercept is \( -16 \).

Final Answer