For the quadratic function defined

Question

For the quadratic function​ defined, ​(a) write the function in the form P(x)=a(x−h)2+k​,

(b) give the vertex of the​ parabola, and ​(c) graph the function.

​P(x)=2x^2-6x-7

a. P(x)=______

B. The vertex is _________

c. Use the graphing tool to graph the function.

Answer 1

Step 1) Move the constant term (7) over to the right to leave a gap and put parenthesis around the first two terms:

P(x) = (2x² - 6x ) - 7

Step 2) Factor out the x² coefficient:

P(x) = 2(x² - 3x ) - 7

Step 3: Add in a new constant term inside the parenthesis that will make the quadratic a perfect square. To find that number, take 1/2 of the coefficient in front of the "x" (in this case that is -3) and then square it (1/2 of -3 is -3/2 and squaring that we get 9/4). Since we are going to add in 9/4 to the quadratic, we are REALLY adding in 2 times 9/4 or 9/2 since we have a "2" on the outside of the parenthesis. To compensate for adding in 9/2, we will also subtract 9/2:

P(x) = 2(x² - 3x + 9/4) - 7 - 9/2

Step 4: Write the quadratic as a square. The number you group with the "x" (inside the square) will be the value you got when you took 1/2 of the coefficient of the "x" term (-3/2). Also, combine the two constant terms:

P(x) = 2(x - 3/2)² - 14/2 - 9/2

P(x) = 2(x - 3/2)² - 23/2

You can then read the vertex right off the equation. The x-coordinate of the vertex is the opposite of the term you grouped with the "x", so 3/2. The y-coordinate of the vertex is the constant term at the end, so -23/2

Vertex: (3/2, -23/2)

You can graph the function with a graphing calculator (like a TI-84) or with an online graphing tool (like desmos)

Answer 2

​a) P(x)=2x^2-6x-7

2(x^2 - 3x) - 7

2(x^2 - 2*1.5*x) - 7

2(x^2 - 2*1.5*x + 1.5^2 - 1.5^2) - 7

2(x^2 - 2*1.5*x + 1.5^2 ) - 2*1.5^2 - 7

2(x- 1.5)^2 - 2*1.5^2 - 7

2(x- 1.5)^2 - 2*2.25 - 7

2(x- 1.5)^2 - 4.5 - 7

2(x- 1.5)^2 - 11.5

b) Vertex 

X = 1.5

Y= -11.5

c) Desmos | Graphing Calculator

Answer 3

2x^2-6x-7=0

2x^2-6x=7

2(x^2-3x)=7

2(x^2-3x+9/4)=23/2

2(x^2-3/2)^2=23/2

2(x^2-3/2)^2-23/2=p(x)

Our vertex will therefore be (3/2,-23/2).

For the graph and additional explanation, feel free to refer to the video.