## Key Concepts

- Define quadratic function.
- Define quadratic parent function.
- Draw the graph of 𝑓(𝑥)=𝑎x
^{2}. - Draw the graph of 𝑓(𝑥)=𝑎x
^{2}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 x^{2}+bx+c = 0

To the factor, a trinomial of the form x^{2}+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) = ax^{2}+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) = x^{2}. It is the simplest function in the quadratic function family.

- The quadratic parent function is f(x)=x
^{2 } - It is the simplest function in the quadratic function family.

### Graph of f(x) = ax^{2}

- 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) = ax^{2} when a<0

- f(x) = ax
^{2}is the reflection of f(x) = -ax^{2}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) = -4x^{2} 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) = 0.15x^{2} 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^{2 }+ 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) = ax
^{2}+ bx + c, where a, b, and c are real numbers and a≠0, is called a quadratic function.

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