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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y-h=a(x-k)^{2}} .
2) Using the standard graphing form, the vertex is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (h,k)} .
3) The $y$-intercept is the point Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (0,y)} where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(0)=y} .

Solution:

Step 1:
First, we put the equation into standard graphing form. Multiplying the equation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y={\frac {1}{3}}x^{2}+2x-3} by 3, we get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3y=x^{2}+6x-9} .

Step 2:
Completing the square, we get Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3y=(x+3)^{2}-18} . Dividing by 3 and subtracting 6 on both sides, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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,
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 18=(x+3)^{2}} . Solving, we get Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=-3\pm 3{\sqrt {2}}} .
Thus, the two $x$ intercepts occur at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-3+3{\sqrt {2}},0)} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-3-3{\sqrt {2}},0)} .

Step 4:
To find the $y$ intercept, we let $x=0$. So, we get $y=-3$.
Thus, the $y$ intercept is (0,-3).

Final Answer:
The vertex is (-3,-6). The equation in standard graphing form is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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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