8A F11 Q3

Question: a) Find the vertex, standard graphing form, and X-intercept for 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 = -3y^2-6y+2}

                  b) Sketch the graph. Provide the focus and directrix.

Note: In this problem, what is referred to as standard graphing form is the vertex form, in case you search on the internet.

Foundations
1) What type of function are we asking you to graph (line, parabola, circle, etc.)?
2) What is the process for transforming the function into the standard graphing form?
3) After we have the standard graphing form how do you find the X-intercept, and vertex?
4) Moving on to part b) How do we find a 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^{rd}} point on the graph?
5) From the standard graphing form how do we obtain relevant information about the focus and directrix?
Answers:
1) The function is a parabola. Some of the hints: We are asked to find the vertex, and directrix. Also only one variable, of x and y, is squared.
2) First we complete the square. Then we divide by the coefficient of x.
3) To find the X-intercept, replace y with 0 and solve for x. Since the parabola is in standard graphing form, the vertex of 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 - h = a(y - k)^2} is (h, k).
4) To find a 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^{rd}} point, we can either use the symmetry of a parabola or plug in another value for x.
5) From the equation 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 - h = a(y - k)^2} , we use the equation 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 a = \frac{1}{4p}} to find p. P is both the distance from the vertex to the focus and the distance from the vertex to the directrix.

Solution:

Step 1:
There are two ways to obtain the standard graphing form.
Regardless of the method the first step is the same: subtract 2 from both sides to yield 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 - 2 = -3y^2 - 6y}
Method 1:
Divide both sides by -3 to make the coefficient of 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^2} , 1. This means 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 \frac{-1}{3}(x - 2) = y^2 + 2y}
Complete the square to 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 \frac{-1}{3}(x - 2) + 1 = (y^2 + 2y + 1) = (y + 1)^2}
Multiply both sides by -3 so 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 - 2) - 3 = -3(y + 1)^2} , and simplify the left side to yield 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 - 5 = -3(y + 1)^2}
Method 2:
Instead of dividing by -3 we factor it out of the right hand side to 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 x - 2= -3(y^2 + 2y)} .
Now we complete the square inside the parenthesis and add -3 to the left hand side resulting in 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 - 5 = -3(y^2 + 2y +1) = -3(y + 1)^2}

Step 2:
Since the parabola is in standard graphing form we can read off the vertex, which is (2, -1).
We get the X-intercept by replacing y with 0 and solving for x. So 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 - 5 = -3(1)^2} , and the X-intercept is (2, 0).

Step 3:
Now we need the value of p using the relation 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 a = \frac{1}{4p}} , where a = -3.
So 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 -3p = \frac{1}{4}} , 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 p = \frac{1}{-12}} .

Step 4:
Since 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 a < 0} , the parabola opens left. Since the focus is inside the parabola, and p tells us the focus is from the vertex, the focus is at 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 (2 - \frac{1}{12}, -1) = (\frac{23}{12}, -1)} .
We also know that the directrix is a vertical line on the outside of the parabola with the distance from the directrix to the vertex being p. Thus the directrix 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 x = 2 + \frac{1}{12} = \frac{25}{2}}

Final Answer:

Vertex: (2, -1), standard graphing form: 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 - 5 = -3(y + 1)^2} , X-intercept: (2, 0), focus: 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 (\frac{23}{12}, -1)} , directrix: 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 = \frac{25}{2}}

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