University:
University of Michigan
Course:
MATH 116 | Calculus II
Academic year:

2024
Views:

132

Pages:

1
Author:

banenetta0l

Question 1 A quadratic function 𝑓 f is given: (a) Express 𝑓 f in standard form. (b) Find the vertex and 𝑥 x- and 𝑦 y-intercepts of 𝑓 f. (c) Sketch the graph. (d) Find the domain and range of 𝑓 f. 𝑓 ( 𝑥 ) = 𝑥 2 − 2 𝑥 + 3 f(x)=x 2 −2x+3 Answer (a) 𝑓 ( 𝑥 ) = 𝑥 2 − 2 𝑥 + 3 f(x)=x 2 −2x+3 The standard form of a quadratic equation is 𝑓 ( 𝑥 ) = 𝑎 ( 𝑥 − ℎ ) 2 + 𝑘 f(x)=a(x−h) 2 +k. 𝑓 ( 𝑥 ) = 𝑥 2 − 2 𝑥 + 1 + 2 ⇒ 𝑓 ( 𝑥 ) = ( 𝑥 − 1 ) 2 + 2 f(x)=x 2 −2x+1+2⇒f(x)=(x−1) 2 +2 On comparing it with the equation 𝑓 ( 𝑥 ) = 𝑎 ( 𝑥 − ℎ ) 2 + 𝑘 f(x)=a(x−h) 2 +k, we get ℎ = 1 , 𝑘 = 2 h=1,k=2. The vertex 𝑉 V is given by: 𝑉 = ( ℎ , 𝑘 ) = ( 1 , 2 ) . V=(h,k)=(1,2). (b) The 𝑦 y-intercept is given by 𝑥 = 0 x=0: 𝑓 ( 0 ) = 3. f(0)=3. So, ( 0 , 3 ) (0,3) is the 𝑦 y-intercept. Now, the 𝑥 x-intercepts are given by equating 𝑓 ( 𝑥 ) = 0 f(x)=0: 𝑥 2 − 2 𝑥 + 3 = 0. x 2 −2x+3=0. 𝑥 2 − 3 𝑥 + 𝑥 − 3 = 0. x 2 −3x+x−3=0. ( 𝑥 − 3 ) ( 𝑥 + 1 ) = 0. (x−3)(x+1)=0. 𝑥 = − 1 , 3. x=−1,3. So, the 𝑥 x-intercepts of 𝑓 f are ( − 1 , 0 ) (−1,0) and ( 3 , 0 ) (3,0). (c) The graph can be sketched as a parabola with vertex ( 1 , 2 ) (1,2), opening upwards. (d) The domain of 𝑓 f is ( − ∞ , ∞ ) (−∞,∞). The range of 𝑓 f is [ 2 , ∞ ) [2,∞).