Use the function f(x) = 2x2 − x − 10 to answer the questions. Part A: Completely factor f(x). (2 points)

Part B: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)

Part C: Describe the end behavior of the graph of f(x). Explain. (2 points)

Part D: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part B and Part C to draw the graph. (4 points)

MAKE IT NEAT

A. x intercepts are where the graph hits the x axis or where f(x)=0

0=2x^2-x-10

solve

hmm

we can use the ac method

multiply 2 and -10 to get -20

what 2 numbers muliply to get -20 and add to get -1 (the coefint of the x term)

-5 and 4

split the midd into that

0=2x^2+4x-5x-10

group

0=(2x^2+4x)+(-5x-10)

factor

0=2x(x+2)-5(x+2)

undistribute

0=(x+2)(2x-5)

set each to 0

0=x+2

0=-2

0=2x-5

5=2x

5/2=x

x intercepts are at x=-2 and 5/2 or the points (-2,0) and (5/2,0)

B. ok, so for f(x)=ax^2+bx+c

if a>0, then the parabola opens up and the vertex is a minimum

if a<0 then the parabola opens down and the vertex is a max

f(x)=2x^2-x-10

2>0

opoens up

vertex is minimum

ok, the vertex

the x value of  the vertex in f(x)=ax^2+bx+c= is -b/(2a)

the y value of the vertex is f(-b/(2a)) so

given

f(x)=2x^2-x-10

a=2

b=-1

-b/2a=-(-1)/(2*2)=1/4

f(1/4)=2(1/4)^2-(1/4)-10

f(1/4)=2(1/16)-1/4-10

f(1/4)=1/8-1/4-10

f(1/4)=1/8-2/8-80/8

f(1/4)=-81/8

so the vertex is (1/4,-81/8) or (0.25,-10.125)

C. graph the x intercepts and the vertex

the vertex is min and the graph goes through the x intercepts

Step-by-step explanation:

A. x = -2, 5/2

B. Minimum

Vertex: (1/4, -81/8)

C. Draw a concave up graph connecting the x-intercepts, vertex and the y-intercept

Step-by-step explanation:

f(x) = 2x² - x - 10

2x² - x - 10 = 0

2x² - 5x + 4x - 10 = 0

x(2x - 5) + 2(2x - 5) = 0

(x + 2)(2x - 5) = 0

x = -2, 5/2

Vertex: (h,k)

h = midpoint of the x-intercepts (due to symmetry)

h = (-2 + 5/2) = 1/4

k = f(1/4)

k = 2(1/4)² - 1/4 - 10 = -81/8

Vertex: (1/4, -81/8)

y-intercept: when x = 0

2(0)² - 0 - 10 = -10

Draw a concave up quadratic connecting thr following points:

(-2,0)

(0,-10)

(1/4,-81/4)

(5/2,0)

a reflection across the x-axis , followed by the by a reflection across the y-axis , and then a scale dilation of 0.5 .