# Advanced Basics Study Guide for the Math Basics

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### A Quadratic Quest

Now we will look at 2nd degree polynomials which are also referred to as quadratics.

#### What is the Meaning of “Quadratic?”

Quadratic comes from the term *quad* which means to square, so the variable in the polynomial is squared. So the expression \(x^2+2x-5\) is a quadratic expression that has a degree of 2.

#### Quadratic Equations

A quadratic equation in the form \(ax^2 + bx + c =0\) where \(a, b, c\) are constants, is said to be in *standard form*.

There are several methods for solving a quadratic equation:

- Completing the square
- Factoring
- Quadratic formula: \(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)
- Graphing

Each of these will find the roots of the quadratic equation.

#### Quadratic Inequalities

Similar to linear inequalities, quadratic inequalities usually have many solutions.

Let’s look at the steps involved in solving a quadratic inequality by factoring.

Solve \(x^2-2x-8 \le 0\)

*Step 1:* First set the quadratic equal to zero and solve for *x* to find the boundary points on the *x*-axis in order to determine the solution.

The points at \(-2\) and \(4\) divide the \(x\) axis into 3 segments.

*Step 2:* Choose an \(x\) value in each of the three segments to determine if the inequality is satisfied.

If \(x = -3\) then:

\[(-3)^2 -2(-3)-8 \lt 0\] \[9+6-8 \lt 0\] \[7 \lt 0 \text{ is not true}\]If \(x = 0\) then:

\[(0)^2 -2(0) - 8 \lt 0\] \[-8 \lt 0 \text{ is true}\]If \(x = 5\) then:

\[(5)^2 -2(5)-8 \lt 0\] \[25-10-8 \lt 0\] \[7 \lt 0 \text{ is not true}\]*Step 3:* Since \(x = 0\) is the only value that makes the inequality true, we shade all numbers on the \(x\) axis between \(-2\) and \(4\), as in our solution set shown in red below.

Stated algebraically, the solution set is \(-2 \le x \le 4\) or \([-2, 4]\) in interval notation.

#### Quadratic Functions

Examples of quadratic functions are:

\[f(x)=x^2-3x+2\] \[v(t) = -5t^2 + 3t -1\] \[H(a) = a^2 -2a\]The *standard form* of a quadratic function is

where \(a, b, c\) are constants.

There are many applications that involve quadratic functions. Revenue generated in selling a product can be stated in the form of a quadratic equation. For example:

Suppose that a company sells \(x\) number of calculators for a price of \(p\) dollars is represented by \(x = 45,000 - 375p\). The revenue, \(R\), generated is the product of the number of calculators sold and the price at which they are sold. Then:

\[R=xp\] \[R = (45,000 - 375p)p\] \[R(p) = -375p^2 + 45,000p\]So, the revenue is a quadratic function that is written in terms of the price.

A quadratic function in *vertex form* is:

where \((h, k)\) is the vertex of the graph of the function, and \(a\) is a vertical stretch of the graph if \(a \gt 1\) or a vertical compression if \(0 \lt a \lt 1\). Also if \(a \lt 0\), then the graph of \(f\) is reflected about the \(x\)-axis.

### The Strange Cases of Non-Linear Graphs

There are a number of graphs that do *not* show a straight line and you may need to at least be able to recognize them. Here are some examples:

#### The graph of a quadratic function, called a *parabola*, is a *U*-shaped curve that has certain attributes you will need to know.

The parent function for the graph of a quadratic function is \(f(x)=x^2\). Its graph is shown below.

Notice the U shape curve for the graph. The vertex is at the origin and represents a turning point where the graph changes direction.

Now let’s look at using the vertex form to transform or move the graph of \(x^2\) around the plane. For example, the graph of \(f(x)=-(x-2)^2 + 3\) has its vertex at \((2, 3)\) and is reflected about the \(x\)-axis since \(a = -1\). The graph is shown below.

#### You will also need to be familiar with higher degree polynomials.

This one, \(f(x)=x^3\), a third-degree polynomial is graphed as an S-shaped curve, shown below.

Like the parabola, its graph can also be transformed in the plane.

#### While not polynomials, you will need to be familiar with other classes of functions such as exponential and logarithmic functions.

Below is the graph of an exponential function \(f(x)=2^x\) and \(g(x) = log_{2}{x}\).

There is a relationship between these two graphs. They are inverses of each other, since reflecting one of the graphs about the line \(y = x\) produces the other graph as shown below.

Again, like the parabola, these graphs can also be transformed in the plane.

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