The University of Winnipeg

  

DEPARTMENT OF
MATHEMATICS & STATISTICS

Precalculus Review Workshop



Quadratic Functions

The graph of a quadratic function is always a parabola. There are two main forms for quadratic functions you should be aware of for your calculus course.
  1. Standard Form: f(x)=ax2+bx+c where a0
    • if a>0, the parabola opens upwards
    • if a<0 the parabola opens downwards
    • This form is useful when you need to factor or use the quadratic formula.
  2. Vertex Form: f(x)=a(xp)2+q where a0
    • if a>0, the parabola opens upwards
    • if a<0 the parabola opens downwards
    • (p,q) is the vertex of the parabola
    • This form is useful when you need to sketch the function or find the vertex of the parabola.
The parabolas of f(x)=a(xp)2+q.

Completing the Square

Given a quadratic function in standard form f(x)=ax2+bx+c, we can find the vertex form by completing the square. Follow the steps below to complete the square.
  1. If a=1 (i.e. f(x)=x2+bx+c):
    1. Calculate (b2)2.
    2. Add and subtract this value to obtain f(x)=x2+bx+(b2)2(b2)2+c.
    3. The function simplifies to f(x)=(x+b2)2b24+c.
  2. If a1:
    1. Factor out the coefficient a from the first two terms: f(x)=a(x2+bax)+c.
    2. Calculate (b2a)2.
    3. Add and subtract this value inside the parentheses to obtain f(x)=a(x2+bax+(b2a)2(b2a)2)+c.
    4. Remove (b2a)2 from the parentheses by multiplying by a: f(x)=a(x2+bax+(b2a)2)a(b2a)2+c.
    5. The function simplifies to f(x)=a(x+b2a)2b24a+c.
Completing the square.

Basic Function Transformations

Below is a chart summarizing transformations of the basic quadratic function f(x)=x2.

Form Description
f(x)+c=x2+c Shift up (c>0) or down (c<0) by c units
f(x+c)=(x+c)2 Shift left (c>0) or right (c<0) by c units
f(x)=x2 Reflection over the x-axis
f(x)=(x)2 Reflection over the y-axis
cf(x)=cx2 Stretch (c>1) or compression (0<c<1) by a factor of c in the y-axis
f(cx)=(cx)2 Stretch (0<c<1) or compression (c>1) by a factor of c in the x-axis

Function transformations.

Problem Set 3.3

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