SOLUTION: Hello, I am having a hard time with the following two problems. Could someone please help with them I would very much appreciate it! Thanks Consider the parabola with equation

Algebra ->  Rational-functions -> SOLUTION: Hello, I am having a hard time with the following two problems. Could someone please help with them I would very much appreciate it! Thanks Consider the parabola with equation       Log On


   

Question 402729: Hello, I am having a hard time with the following two problems. Could someone please help with them I would very much appreciate it! Thanks
Consider the parabola with equation f(x)= -x^2-2x+15
(a) Does the parabola open up or down?
(b) What is the equation for the axis of symmetry for the parabola?
(c) What are the coordinates of the vertex of the parabola?
(d) What is the y-intercept of the parabola?
(e) Determine the x-intercepts if there are any.
(f) Sketch a graph of the parabola.

A parabola has an x-intercept at 2, its axis of symmetry is the line x = 4, and the y- coordinate of its vertex is 6. Determine the equation of the parabola. Hint: The equation looks like y = ax^2+bx+c. Use the given information to figure out what a, b, c must be. For example, when x = 2, the y value must be 0 since 2 is an x-intercept. That means 0 = 4a + 2b + c. The other two given facts provide similar information about a, b, c.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Consider the parabola with equation f(x)= -x^2-2x+15
(a) Does the parabola open up or down?
Down, the x^2 term is negative.
---------------
(b) What is the equation for the axis of symmetry for the parabola?
x = -b/2a = 2/(-2)
x = -1
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(c) What are the coordinates of the vertex of the parabola?
It's (x,f(x)) on the line of symmetry.
f(-1) = 16
--> (-1,16)
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(d) What is the y-intercept of the parabola?
f(0) = 15 or (0,15)
--------------
(e) Determine the x-intercepts if there are any.
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case -1x%5E2%2B-2x%2B15+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2%29%5E2-4%2A-1%2A15=64.

Discriminant d=64 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+64+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+64+%29%29%2F2%5C-1+=+-5
x%5B2%5D+=+%28-%28-2%29-sqrt%28+64+%29%29%2F2%5C-1+=+3

Quadratic expression -1x%5E2%2B-2x%2B15 can be factored:
-1x%5E2%2B-2x%2B15+=+%28x--5%29%2A%28x-3%29
Again, the answer is: -5, 3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B-2%2Ax%2B15+%29

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(f) Sketch a graph of the parabola.
dl the FREE graph software at
http://www.padowan.dk