Questions: Consider the following function. Complete parts (a) through (e) below. f(x)=-x^2+6x-5 (Type an integer or a fraction.) e. Use the results from parts (a)-(d) to graph the quadratic function. Use the graphing tool to graph the function. Click to enlarge graph

Solution Steps

The given quadratic function is \( f(x) = -x^2 + 6x - 5 \). To find the vertex, we use the vertex formula \( x = -\frac{b}{2a} \).

Here, \( a = -1 \) and \( b = 6 \).

\[ x = -\frac{6}{2(-1)} = 3 \]

Now, substitute \( x = 3 \) back into the function to find \( y \):

\[ f(3) = -(3)^2 + 6(3) - 5 = -9 + 18 - 5 = 4 \]

So, the vertex is \( (3, 4) \).

The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \).

From Step 1, we already calculated this:

\[ x = 3 \]

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

\[ f(0) = -0^2 + 6(0) - 5 = -5 \]

So, the y-intercept is \( (0, -5) \).

Final Answer