004 Sample Final A, Problem 2

a) Find the vertex, standard graphing form, and x-intercepts for $f(x)={\frac {1}{3}}x^{2}+2x-3$
b) Sketch the graph. Provide the y-intercept.

Foundations
1) What is the standard graphing form of a parabola?
2) What is the vertex of a parabola?
3) What is the $y$ -intercept?
Answer:
1) Standard graphing form is $y-h=a(x-k)^{2}$ .
2) Using the standard graphing form, the vertex is $(h,k)$ .
3) The $y$ -intercept is the point $(0,y)$ where $f(0)=y$ .

Solution:

Step 1:
First, we put the equation into standard graphing form. Multiplying the equation $y={\frac {1}{3}}x^{2}+2x-3$ by 3, we get
$3y=x^{2}+6x-9$ .

Step 2:
Completing the square, we get $3y=(x+3)^{2}-18$ . Dividing by 3 and subtracting 6 on both sides, we have
$y+6={\frac {1}{3}}(x+3)^{2}$ .

Step 3:
From standard graphing form, we see that the vertex is (-3,-6). Also, to find the $x$ intercept, we let $y=0$ . So,
$18=(x+3)^{2}$ . Solving, we get $x=-3\pm 3{\sqrt {2}}$ .
Thus, the two $x$ intercepts occur at $(-3+3{\sqrt {2}},0)$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-3-3\sqrt{2},0)} .

Step 4:
To find the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} intercept, we let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} . So, we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=-3} .
Thus, the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} intercept is (0,-3).

Final Answer:
The vertex is (-3,-6). The equation in standard graphing form is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y+6=\frac{1}{3}(x+3)^2} .
The two Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} intercepts are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-3+3\sqrt{2},0)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-3-3\sqrt{2},0)} .
The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} intercept is (0,-3)

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