Question

g(x) = -x² + 4x - 5
(a) write the equation in the form g(x) = (x - h)² + k or the form g(x) = -(x - h)² + k.
g(x) =
(b) find the equation of the axis of symmetry.
equation of axis of symmetry:
(c) does the function have a minimum or maximum value (output)?
minimum maximum
(d) for what input, x, does the minimum or maximum value occur?
x =
(e) what is the function’s minimum or maximum value?

Explanation:

Step1: Complete the square for part (a)

Given \( g(x) = -x^2 + 4x - 5 \), factor out the coefficient of \( x^2 \) from the first two terms: \( g(x) = - (x^2 - 4x) - 5 \). To complete the square inside the parentheses, take half of -4 (which is -2), square it (which is 4), and add and subtract it inside the parentheses. But since there's a factor of -1 outside, we have: \( g(x) = - (x^2 - 4x + 4 - 4) - 5 = - ((x - 2)^2 - 4) - 5 = - (x - 2)^2 + 4 - 5 = - (x - 2)^2 - 1 \).

Step2: Find axis of symmetry for part (b)

For a quadratic function in the form \( g(x) = a(x - h)^2 + k \), the axis of symmetry is \( x = h \). From part (a), \( h = 2 \), so the axis of symmetry is \( x = 2 \).

Step3: Determine min or max for part (c)

The coefficient of \( x^2 \) is -1, which is negative. So the parabola opens downward, meaning the function has a maximum value.

Step4: Find input for max/min for part (d)

For a quadratic function in vertex form \( g(x) = a(x - h)^2 + k \), the vertex is at \( (h, k) \). The maximum (or minimum) occurs at \( x = h \). From part (a), \( h = 2 \), so \( x = 2 \).

Step5: Find max/min value for part (e)

From the vertex form \( g(x) = - (x - 2)^2 - 1 \), the vertex is \( (2, -1) \). Since the parabola opens downward, the maximum value is \( k = -1 \).

Answer:

s:
(a) \( g(x) = \boldsymbol{ - (x - 2)^2 - 1 } \)
(b) Equation of axis of symmetry: \( \boldsymbol{ x = 2 } \)
(c) The function has a \(\boldsymbol{\text{Maximum}}\) value.
(d) \( x = \boldsymbol{ 2 } \)
(e) The function's maximum value is \(\boldsymbol{ -1 }\)