# Advanced Basics Study Guide for the Math Basics

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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.

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:

1. Completing the square
2. Factoring
3. Quadratic formula: $$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$$
4. Graphing

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

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.

$x^2-2x-8 = 0$ $(x+2)(x-4)=0$ $x+2 = 0 \text{ or } x-4 = 0$ $x = -2 \text{ or } x = 4$

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.

$f(x)=x^2-3x+2$ $v(t) = -5t^2 + 3t -1$ $H(a) = a^2 -2a$

The standard form of a quadratic function is

$f(x)=ax^2+bx +c$

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:

$f(x) = a(x-h)^2 + k$

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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