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# Key Features of Quadratic Function

## Key Concepts

• 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) = ax+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.

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.

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.

### 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) = 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

### 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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