Answer to Question #130449 in Algebra for Everlyn

Question

Use the graph to find the following.
(a) the domain of f
(b) the range of f  
(c) the x-intercepts
(d) the y-intercept
(e) intervals on which f is increasing
(f) intervals on which f is decreasing
(g) intervals on which f is constant
(h) the number at which f has a relative minimum
(i) the relative minimum of f   
(j) f(0)(k) The values of x for which f(x)=3(l) Is f even, odd or neither?
The 1st coordinate is -6,24. The 2nd coordinate is 0,24.The 3rd coordinate is 5,-1.The 4th coordinate is 10,26.

Answer 1

The function is

$$f(x)= \begin{cases} 24 & x<0 \\ ax^2+bx+c &x\geq0 \end{cases}$$

$f(0)=24: 24=c$

$f(5)=-1: -1=a(5)^2+b(5)+24$

$f(10)=26: 26=a(10)^2+b(10)+24$

$\begin{matrix} 25a+5b=-25 \\ 100a+10b=2 \end{matrix}$

$\begin{matrix} b=-5a-5 \\ 25a=26 \end{matrix}$

$\begin{matrix} a=1.04 \\ b=-10.2 \end{matrix}$

$$f(x)= \begin{cases} 24 & x<0 \\ 1.04x^2-10.2x+c &x\geq0 \end{cases}$$

$x_{vertex}=-\dfrac{b}{2a}=-\dfrac{-10.2}{2(1.04)}=\dfrac{255}{52}$

$y_{vertex}=1.04(\dfrac{255}{52})^2-10.2(\dfrac{255}{52})+24=-\dfrac{105}{104}$

$f(0)=24$

$$y=0: 1.04x^2-10.2x+24=0$$

$$1.04x^2-10.2x+24=0$$

$$26x^2-255x+600=0$$

$$x=\dfrac{255\pm\sqrt{(-255)^2-4(26)(600)}}{2(26)}=\dfrac{255\pm5\sqrt{105}}{52}$$

$$x_1=\dfrac{255-5\sqrt{105}}{52}, x_2=\dfrac{255+5\sqrt{105}}{52}$$

$$f(x)=3=1.04x^2-10.2x+24$$

$$1.04x^2-10.2x+21=0$$

$$26x^2-255x+525=0$$

$$x=\dfrac{255\pm\sqrt{(-255)^2-4(26)(525)}}{2(26)}=\dfrac{255\pm5\sqrt{417}}{52}$$

$$x_1=\dfrac{255-5\sqrt{417}}{52}, x_2=\dfrac{255+5\sqrt{417}}{52}$$

(a) Domain: $(-\infin,\infin)$

(b) Range: $( -\dfrac{105}{104}, \infin)$

(c) the x-intercepts

$(\dfrac{255-5\sqrt{105}}{52}, 0), (\dfrac{255+5\sqrt{105}}{52}, 0)$

(d) the y-intercept

$(0,24)$

(e) intervals on which f is increasing

$(\dfrac{255}{52}, \infin)$

(f) intervals on which f is decreasing

$(0, \dfrac{255}{52})$

(g) intervals on which f is constant

$(-\infin, 0)$

(h) the number at which f has a relative minimum

$\dfrac{255}{52}$

(i) the relative minimum of f

$-\dfrac{105}{104}$

(j) f(0)

$24$

(k) The values of x for which f(x)=3

$\dfrac{255-5\sqrt{417}}{52}, \dfrac{255+5\sqrt{417}}{52}$

(l) Is f even, odd or neither?

The function f is neither even nor odd.