Key Features of Quadratic Function  

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.  

Transformations of exponential functions 

Vertical translation of graphs 

Algebra: f(x) = a^x+k

Horizontal translation of graphs 

Algebra: f(x) = a^(x−h)

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. 

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.  

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. 

Graph of a quadratic equation 

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

• 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)=x² 
• It is the simplest function in the quadratic function family. 

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 when a<0

• f(x) = ax² is the reflection of f(x) = -ax² over the x-axis. 

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?

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

Concept Map

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. 

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

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.