How do you graph the quadratic function and identify the vertex and axis of symmetry and x intercepts for $y = \frac{1}{3} (x + 4) (x + 1)$ $y=1/3(x+4)(x+1)$ ?

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How do you graph the quadratic function and identify the vertex and axis of symmetry and x intercepts for $y = \frac{1}{3} (x + 4) (x + 1)$ $y=1/3(x+4)(x+1)$ ?

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Answer 1 · 2018-07-24T04:43:43

The vertex is $(- \frac{5}{2}, - \frac{3}{4})$ $(-5/2,-3/4)$ or $(- 2.5, - 0.75)$ $(-2.5,-0.75)$ .
The y-intercept is $(0, \frac{4}{3})$ $(0,4/3)$ or $(0, \approx 1.333)$ $(0,~~1.333)$ .
The x-intercepts are $(- 1, 0), (- 4, 0)$ $(-1,0), (-4,0)$ .
Additional point: $(- 5, \frac{4}{3})$ $(-5,4/3)$ or $(- 5, \approx 1.333)$ $(-5,~~1.333)$ .

Explanation:

Given:

$y = \frac{1}{3} (x + 4) (x + 1)$ $y=1/3(x+4)(x+1)$

Expand $(x + 4) (x + 1)$ $(x+4)(x+1)$ .

$y = \frac{1}{3} (x^{2} + 5 x + 4)$ $y=1/3(x^2+5x+4)$

Distribute $\frac{1}{3}$ $1/3$ .

$y = \frac{1}{3} x^{2} + \frac{5}{3} x + \frac{4}{3}$ $y=1/3x^2+5/3x+4/3$ is a quadratic equation in standard form:

$y = a x^{2} + b x + c$ $y=ax^2+bx+c$ ,

where:

$a = \frac{1}{3}$ $a=1/3$ , $b = \frac{5}{3}$ $b=5/3$ , and $c = \frac{4}{3}$ $c=4/3$

The vertex is the maximum or minimum point of the parabola. The formula for the axis of symmetry gives us the x-coordinate of the vertex:

$x = \frac{- b}{2 a}$ $x=(-b)/(2a)$

$x = \frac{- \frac{5}{3}}{2 \cdot \frac{1}{3}}$ $x=(-5/3)/(2*1/3)$

$x = \frac{- \frac{5}{3}}{\frac{2}{3}}$ $x=(-5/3)/(2/3)$

$x = - \frac{5}{3} \times \frac{3}{2}$ $x=-5/3xx3/2$

$x = - \frac{15}{6}$ $x=-15/6$

$x = - \frac{5}{2}$ $x=-5/2$ or $2.5$ $2.5$

To find the y-coordinate of the vertex, substitute $- \frac{5}{2}$ $-5/2$ for $x$ $x$ and solve for $y$ $y$ .

$y = \frac{1}{3} {(- \frac{5}{2})}^{2} + \frac{5}{3} (- \frac{5}{2}) + \frac{4}{3}$ $y=1/3(-5/2)^2+5/3(-5/2)+4/3$

$y = \frac{1}{3} (\frac{25}{4}) - \frac{25}{6} + \frac{4}{3}$ $y=1/3(25/4)-25/6+4/3$

$y = \frac{25}{12} - \frac{25}{6} + \frac{4}{3}$ $y=25/12-25/6+4/3$

The least common denominator is $12$ $12$ . Multiply $\frac{25}{6} \times \frac{2}{2}$ $25/6xx2/2$ and $\frac{4}{3} \times \frac{4}{4}$ $4/3xx4/4$ to get equivalent fractions. Since $\frac{n}{n} = 1$ $n/n=1$ , the numbers will change but the value of each fraction will not change.

$y = \frac{25}{12} - \frac{25}{6} \times \frac{2}{2} + \frac{4}{3} \times \frac{4}{4}$ $y=25/12-25/6xxcolor(red)2/color(red)2+4/3xxcolor(blue)4/color(blue)4$

$y = \frac{25}{12} - \frac{50}{12} + \frac{16}{12}$ $y=25/12-50/12+16/12$

$y = - \frac{9}{12}$ $y=-9/12$

$y = - \frac{3}{4}$ $y=-3/4$ or $- 0.75$ $-0.75$

The vertex is $(- \frac{5}{2}, - \frac{3}{4})$ $(-5/2,-3/4)$ or $(- 2.5, - 0.75)$ $(-2.5,-0.75)$ . Plot this point.

The y-intercept is the value of $y$ $y$ when $x = 0$ $x=0$ . Substitute $0$ $0$ for $x$ $x$ and solve for $y$ $y$ .

$y = \frac{1}{3} {(0)}^{2} + \frac{5}{3} (0) + \frac{4}{3}$ $y=1/3(0)^2+5/3(0)+4/3$

$y = \frac{4}{3}$ $y=4/3$ or $\approx 1.333$ $~~1.333$

The y-intercept is $(0, \frac{4}{3})$ $(0,4/3)$ or $(0, \approx 1.333)$ $(0,~~1.333)$ . Plot this point.

The x-intercepts are the values for $x$ $x$ when $y = 0$ $y=0$ . Substitute $0$ $0$ for $y$ $y$ and solve for $x$ $x$ .

$0 = \frac{1}{3} x^{2} + \frac{5}{3} x + \frac{4}{3}$ $0=1/3x^2+5/3x+4/3$

Switch sides.

$\frac{1}{3} x^{2} + \frac{5}{3} x + \frac{4}{3} = 0$ $1/3x^2+5/3x+4/3=0$

Multiply both sides by $3$ $3$ .

$x^{2} + 5 x + 4 = 0$ $x^2+5x+4=0$

Factor $x^{2} + 5 x + 4$ $x^2+5x+4$ .

$(x + 1) (x + 4) = 0$ $(x+1)(x+4)=0$

Set each binomial to zero and solve.

$x + 1 = 0$ $x+1=0$

$x = - 1$ $x=-1$

$x + 4 = 0$ $x+4=0$

$x = - 4$ $x=-4$

The x-intercepts are $(- 1, 0), (- 4, 0)$ $(-1,0), (-4,0)$ . Plot these points.

Additional point: $x = - 5$ $x=-5$

$x = - 5$ $x=-5$ is the mirror of the x-coordinate of the y-intercept.

Substitute $- 5$ $-5$ for $x$ $x$ and solve for $y$ $y$ .

$y = \frac{1}{3} {(- 5)}^{2} + \frac{5}{3} (- 5) + \frac{4}{3}$ $y=1/3(-5)^2+5/3(-5)+4/3$

$y = \frac{25}{3} - \frac{25}{3} + \frac{4}{3}$ $y=25/3-25/3+4/3$

Additional point: $(- 5, \frac{4}{3})$ $(-5,4/3)$ or $(- 5, \approx 1.333)$ $(-5,~~1.333)$ . Plot this point.

Sketch a graph through the points. Do not connect the dots.

graph{y=(x^2)/3+(5x)/3+4/3 [-10, 10, -5, 5]}

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Explanation:

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