Explained: "Consider the quadratic function f(x)=−2x2+6x−2. A) what are..."

High School
Algebra
Algebra I
Quadratic function and equation

jeliceg2

Answered question

2021-12-03

Consider the quadratic function

f (x) = - 2 x^{2} + 6 x - 2

.
A) what are the y and x incerpts. Use these intercepts to write the function in factored form.
B)Use completing the square to write the function in standard form.
C) list a sequence of transformations to get from the graph of

y = x^{2}

to the graph of f. Sketch a graph of

y = f (x)

.
Does the graph open upward or downward. Where does the global maximun or minimun value of the function occur. What are thye global max. or min value. where is the function increasing and decreasing?
D) What are the domain and range of the function? Giv ethe equation of the parabolas axis of symmetry.

Answer & Explanation

Alfonso Miller

Beginner2021-12-04Added 20 answers

A) y-intercepts: Here,

x = 0

, so, we get

f (0) = - 2

. So, the y-intercept is(0,-2).
x-intercepts: Here,

y = f (x) = 0

,so, we get

- 2 x^{2} + 6 x - 2 = 0

\Rightarrow x = \frac{- 6 \pm \sqrt{36 - 4 (- 2) (- 2)}}{- 4}

\Rightarrow x = \frac{- 6 \pm \sqrt{36 - 16}}{- 4}

\Rightarrow x = \frac{- 6 \pm \sqrt{20}}{- 4}

\Rightarrow x = \frac{- 6 \pm 2 \sqrt{5}}{- 4}

\Rightarrow x = \frac{3 \pm \sqrt{5}}{2}

Thus, the function in factored form is

f (\begin{matrix} x \end{matrix}) = - 2 (\begin{matrix} x - \frac{3 + \sqrt{5}}{2} \end{matrix}) (\begin{matrix} x - \frac{3 - \sqrt{5}}{2} \end{matrix})

.
B)
The square completion can be done as follows:

f (x) = - 2 (x^{2} - 3 x + 1)

[Take -2 common to make leading coefficient unit]

\Rightarrow f (x) = - 2 (\begin{matrix} x^{2} - 3 x + (\begin{matrix} {\frac{- 3}{2}}^{2} \end{matrix}) - (\begin{matrix} {\frac{- 3}{2}}^{2} \end{matrix}) + 1 \end{matrix})

[Add and subtract square of half the coefficient of x]

\Rightarrow f (x) = - 2 (\begin{matrix} (\begin{matrix} x - \frac{3}{2} \end{matrix})^{2} - \frac{9}{4} + 1 \end{matrix})

\Rightarrow f (x) = - 2 (\begin{matrix} (\begin{matrix} x - \frac{3}{2} \end{matrix})^{2} - \frac{5}{4} \end{matrix})

\Rightarrow f (x) = - 2 (\begin{matrix} x - \frac{3}{2} \end{matrix})^{2} + \frac{5}{2}

C)
The function is obtained by the following transformation to

y = x^{2}

:
1. Horizontal shift to the right by

\frac{3}{2}

units.
2. Vertical shift upwards by

\frac{5}{2}

units.
3. Vertical stretching by a factor of 2.
4. Reflecting the function along x-axis.
The graph is as follows:
The graph opens downwards. The global maximum occurs at x=1.5. The global maximum value is

f (1.5) = 2.5

.
The function is increasing on the interval

(- \infty, 1.5)

while itis decreasing on the interval

(1.5, + \infty)

.
D)

The domain of the function is set of all real numbers, R. The range of the function is the interval

(- \infty, 1.5] = {y : - \infty < y \leq 5 and y \in R}

.
The axis of symmetry has the equation:

x = 1.5

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