Question

For the function below, (a) find the vertex; (b) find the axis of symmetry: (c) determine a minimum value and find that value; and (d) graph the function. f(x) = x2 + 8x + 17 (a) The vertex is (Type an ordered pair, using integers or fractions.) (b) The axis of symmetry is (Type an equation. Use integers or fractions for any numbers in the equation.) (c) Does f(x) have a maximum or a minimum value? The parabola f(x) has a minimum value. 0 The parabola f(x) has a maximum value. The minimum/maximum value is (Type an integer or a fraction.) (d) Use the graphing tool to graph the function. Click to enlarge graph

Answer 1

$\small f(x) =x^{2}+8x+17$

Covert the quadratic function f(x) in vertex form, i.e, $\small f(x) =(x-h)^{2}+k$

$\small f(x) =x^{2}+8x+16-16+17$

$\small f(x) =(x^{2}+8x+16)-16+17$

$\small f(x) =(x+4)^{2}+1$

a) Vertex = (h, k) = (-4, 1)

b) Quadratic function f(x) is symmetric about x = -4.

c)

Differentiate f(x) with respect to x,

$\small f'(x) = 2x+8$

Put f'(x) = 0 to find critical points,

$\small f'(x) = 2x+8=0\Rightarrow x = -4$

Also,

$\small f''(x) = 2>0$,

Therefore, f(x) has only minima at x = -4

$\small f(-4) =(-4)^{2}+8(-4)+17=16-32+17=1$

Minimum value of f(x) = 1

d)