Solving & Graphingthe Quadratic Equation

Created by Yvette Lee Source: mathisfun, purplemath, www.chaoticgolf.com/pptlessons/graphquadraticfcns2.ppt

What is quadratic equa-tion?

The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2)

Quadratic Functions

The graph of a quadratic function is a parabola.

A parabola graph has an open U-shape

NOTE! Make sure the parabola doesn’t stop at the end of the curve. It is continuous for all x-values.

y

x

y

x

Line of Symmetry

Line of SymmetryParabolas have a symmetric property to them.

If we drew a line down the mid-dle of the parabola, we could fold the parabola in half.

We call this line the line of symmetry.

The line of symmetry ALWAYS passes through the middle point hor-

izontally.

Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.

What is the  Standard Form of a Quadratic Equation?

* a, b and c are known values. a can't be 0.* "x" is the variable or unknown (you don't know it yet).

Let’s find a, b, and c in the examples below.

In this one a=2, b=5 and c=3

Where is a?

 In fact a=1, as we don't usually write "1x2"

b = -3 And where is c?Well, c=0, so is not shown.

Oops! This one is not a quadratic equation, because because it is missing x2 (in other words a=0, and that means it can't be quadratic)

 

But sometimes a quadratic equation doesn't look like that! For example:

How do we find solutions to the quadratic equations?

What is solutions?

The "solutions" to the Quadratic Equation are where it is equal to zero. There are usually 2 solutions (as shown in the graph above).

They are also called "roots", or sometimes "zeros"

How to graph the quadratic equa-tion

1. Find the Standard form of the quadratic equation.

2. Find the solutions(zeroes, roots) of the equation. They are x-values on the x-axis 3. Find y-intercept (c value in the standard form, or the term without variables)

S -> X -> Y Standard form x-value (roots) y-intercept

A trick when you see (x-a)(x-b)=0Solve (x + 1)(x – 3) = 0

This is a quadratic, and I'm supposed to solve it. I could multiply the left-hand side, simplify to find the coefficients, plug them into the Quadratic Formula, and chug away to the answer.

But why would I? I mean, for heaven's sake, this is factorable, and they've already factored it and set it equal to zero for me. While the Quadratic For-mula would give me the correct answer, why bother with it? Instead, I'll just solve the factors:

(x + 1)(x – 3) = 0 x + 1 = 0  or  x – 3 = 0 x = –1  or  x = 3The solution is  x = –1, 3

* If you want to distribute the equation and use the formula to find the roots, it works too. This is just a easier and quicker way to solve it.

Solve and graph x2+5x=-6S -> X -> Y

1. Is it the standard form?

No! I add 6 in each side to make the right side to zero: x2+5x+6=0

2. What are the roots/zeroes/solutions?

a=1, b=5, c=6Plug the numbers in the calculator or use factoring (x+2)(x+3)=0

The solution is x=2, or 3

3. Find y-intercept. C=6, so y-intercept is 6.

Time to draw the graph. Plot the roots and the y-intercept and make a symmetrical U shape

Solve and graph x2+4x=-4S -> X -> Y

1. Is it the standard form?

No! I add 4 in each side to make the right side to zero: x2+4x+4=0

2. What are the roots/zeroes/solutions?

a=1, b=4, c=4Plug the numbers in the calculator or use factoring (x+2)(x+2)=0

The solution is x=2, or 2

3. Find y-intercept. C=4, so y-intercept is 4.

Time to draw the graph. Plot the roots and the y-intercept and make a symmetrical U shape