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Key Features of Quadratic Function  

Sep 17, 2022
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Key Concepts

  • Define quadratic function.
  • Define quadratic parent function.
  • Draw the graph of 𝑓(𝑥)=𝑎x2.
  • Draw the graph of 𝑓(𝑥)=𝑎x2 when 𝑎<0.
  • Interpret quadratic functions from the table.
  • Compare the rate of change on the graph.

Exponential function 

An exponential function is the product of an initial amount and a constant ratio raised to a power.  

Exponential function 

Transformations of exponential functions 

Vertical translation of graphs 

Algebra: f(x) = ax+k

Vertical translation of graphs 

Horizontal translation of graphs 

Algebra: f(x) = a(x−h)

Horizontal translation of graphs 

Polynomial 

A polynomial expression is an expression that has constants and variables by means of addition, multiplication, and exponentiation to a non-negative integer power. 

Polynomial 

Factorizing x2+bx+c = 0 

To the factor, a trinomial of the form x2+bx+c, find a factor pair of c that has a sum of b. Then use the factors you found to write the binomials that have a product equal to the trinomial.  

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Quadratic function 

A function f defined by  f(x) = ax2+bx+c, where  a, b and c are real numbers and  a≠0a≠0, is called a quadratic function

Quadratic function 

Graph of a quadratic equation 

The graph of a quadratic function is a curve called a parabola. 

Graph of a quadratic equation
  • The axis of symmetry intersects the vertex and divides the parabola in half.  
  • The vertex is the lowest (or highest) point on the graph of a quadratic function. 

Quadratic parent function 

The quadratic parent function is f(x) = x2. It is the simplest function in the quadratic function family. 

  • The quadratic parent function is f(x)=x
  • It is the simplest function in the quadratic function family. 
Quadratic parent function 

Graph of f(x) = ax2

  • For 0<|a|<1, the shape of the parabola is wider than the parent function. 
  • For |a|>1, the shape of the parabola is narrower than the parent function. 
Graph of f(x) = ax2

Graph of f(x) = ax2 when a<0

  • f(x) = ax2 is the reflection of f(x) = -ax2 over the x-axis. 
Graph of f(x) = ax2 when a<0

Compare the rate of change in the graph 

  • Find the slope of the line that passes through each pair of points. 
  • For positive intervals, the greater the value of a the greater the average rate of change. In this case, the ratio of the a-values in the two functions is the same as the ratio of the average rates of change. 

Exercise

1. How does the value of a in g(x) = -4x2 affect the graph when compared to the graph of the quadratic parent function?

2. In which interval is the function increasing?

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exercise 2

3. How does the value of a in h(x) = 0.15x2 affect the graph when compared to the graph of the quadratic parent function?

4. Write a quadratic equation for the area of the figure given. Find the area of the figure for the given value of x.

x=5

exercise 4

Concept Map

A function 𝑓 defined by 𝑓(𝑥) = 𝑎𝑥+ 𝑏𝑥 + 𝑐, where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0, is called a quadratic function

The graph of a quadratic function is a curve called a parabola

Concept Map

What we have learned

  • A function f defined by f(x) = ax2 + bx + c, where a, b, and c are real numbers and a≠0, is called a quadratic function.

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