Question

Two Truths and a Lie: Remember, the function has a maximum and an axis of symmetry at x = 2. Two of the following functions could represent this scenario while one cannot. Determine which two could represent the scenario and which one cannot. Explain how you know and show all necessary work to support your reasoning. a) f(x) = −x 2 + 4x + 2 b) g(x) = −(x + 2) 2 − 5 c) h(x) = −2(x − 1)(x − 3)

Answer 1

The functions f(x) and h(x) have an axis of symmetry of x = 2 and a maximum

How to determine the function?

The functions are given as:

• f(x) = -x² + 4x + 2
• g(x) = -(x + 2)² - 5
• h(x) = -2(x − 1)(x − 3)

A quadratic function can be represented as:

y = ax² + bx + c

y = a(x - h)² + k

In both cases, when a < 0; the function has a maximum.

This means that all three functions have a maximum.

Next, we determine the axis of symmetry:

• In y = ax² + bx + c, the axis of symmetry is x = -b/2a
• In y = a(x - h)² + k, the axis of symmetry is x = h

Using the above forms, we have:

f(x) = -x² + 4x + 2

Axis of symmetry, x = -4/(2 * -1)

x = 2

g(x) = -(x + 2)² - 5

Axis of symmetry, x = -2

h(x) = -2(x − 1)(x − 3)

Expand

h(x) = -2x² - 8x + 8

Axis of symmetry, x = -8/(2 * -2)

x = 2

This means that f(x) and h(x) have an axis of symmetry of x = 2

Hence, the functions f(x) and h(x) have an axis of symmetry of x = 2 and a maximum

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