Chapter 10. In this chapter we will examine quadratic graphs and their equations. We will solve quadratic equations by various techniques such as

Chapter 10

Quadratic Equations

and Functions

In this chapter we will examine quadratic graphs and their equations.

We will solve quadratic equations by various techniques such as factoring, finding the square roots, completing the square, and applying the quadratic formula.

We will also learn about the discriminant and how it is used to characterize the roots of a quadratic equation.

Introduction

Examples: y = 5x2; y = x2 + 7; y = x2 – x – 3The graph of a quadratic function is a U-

shaped curve known as a parabola. A parabola has an axis of symmetry, which is an

imaginary line that divides the parabola into two identical halves.

Vertex: The highest or lowest point of a parabola and is located on the axis of symmetry.

The vertex can be determined from a graph or from a equation.

Exploring Quadratic Graphs (10.1)

Standard Form of a Quadratic FunctionA quadratic function is a function that can be written in the form y = ax2 + bx + c, where a ≠ 0. This form is called the standard form of a quadratic function.


If a>0 in y = ax2 + bx + c, then the parabola opens upward and the vertex is the minimum point or lowest point of the parabola.

If a<0 in y = ax2 + bx + c, then the parabola opens downward and the vertex is the maximum point or highest point of the parabola.

We can use the fact that a parabola is symmetric to graph it quickly. Evaluate the quadratic function to find the

coordinates of the vertex and several points on either side of the vertex. Using a table will be helpful here.

Then reflect the points across the axis of symmetry. For quadratic functions of the form y = ax2, the vertex

is at the origin.


The value of a, the coefficient of the x2 term in the quadratic function, affects the width .

From the examples, we see that for the graph of y = mx2 is wider than the graph of y = nx2. Thus the larger the value of a, the more narrow the

parabola.


The sign associated a, the coefficient in the quadratic function, will determine the direction the parabola will open up. A positive value of a will direct the parabola to open

up. A negative value for a will direct the parabola to open

down.


The value of c, the constant term in the quadratic function, translates the graph up or down. A positive value of c will translate the parabola up. A negative value for c will translate the parabola

down.


We can model the height of an object moving under the influence of gravity using a quadratic function. The motion is known as a free fall and if we trace the

path of a free falling object it will trace out a parabola. In free fall an object’s speed continues to increase. If we ignore air friction we can find the approximate

height of a falling object using the function: h = -16t2 + c (h=final height, t=time, c=initial height).


Sample Problem Suppose you see an eagle flying over a canyon. The eagle is 30 ft above the level of the canyon’s edge when it drops a stick from its claws. The force of gravity causes the stick to fall toward Earth. Graph this motion.


Remember, the axis of symmetry for the quadratic function y = ax2 + c is the y-axis (Section 10.1). The value of a affects the direction the parabola points

and how wide it will be. The value for c translates the parabola up or down.

For the quadratic function y = ax2 + b + c, the value of b affects the position of the axis of symmetry.

Quadratic Functions (10.2)


Graph of a Quadratic EquationThe graph of y = ax2 + bx + c, where a ≠ 0, has the line x = as its axis of symmetry. the x-coordinate of the vertex is .

Examples:

When we substitute x = 0 into the equation y = ax2 + bx + c, y = c. Therefore, the y-intercept of a quadratic function is

the value of c.We can use the axis of symmetry and the

y-intercept to help us graph a quadratic function.


Sample ProblemGraph the function y = -3x2 + 6x + 5

Graphing quadratic inequalities is similar to graphing linear inequalities. The curve will be a dashed line if the inequality

involves a < or >. The curve is a solid if the inequality involves . If an inequality is written in terms of y < or y , shade

the region below the boundary (or the region outside the curve).

If an inequality is written in terms of y > or y , shade the region above the boundary (or the region inside the curve).


Every positive number ha two square roots.A radical symbol () indicates a square root.

The expression means the positive, or principal square root.

The expression means the negative square root. The expression under the radical sign is known as the

radicand.

Finding and Estimating Square Roots (10.3)

Square RootThe number a is a square root of b if a2 = b.

Sample ProblemSimplify each expression.

Some square roots are rational numbers and some are irrational numbers. Example: Rational roots Example: Irrational roots

Remember: In decimal form a rational number terminates or repeats, whereas an irrational number continues without repeating.


The squares of integers are called perfect squares. Consecutive integers: {1, 2, 3, 4, 5, 6} Consecutive perfect squares: {1,4,9,16,25,36}

We can use perfect squares to estimate square root values.


Sample Problem Between what two consecutive integers is ?

We can apply square roots to real world situations. Problem solving involving square roots.


Sample Problem The formula gives the length d of each wire for the tower at the right. Find the length of the wire if x = 12 ft.

d

x

2x

A quadratic equation can have two, one, or no real-number solutions. In future courses you will learn about solutions of

quadratic equations that are not real numbers. In this course, solutions will refer to real-number solutions.

There is a relationship between the solution of a quadratic equation and its related quadratic function (y = ax2 + bx + c): The solutions of a quadratic equation and the x-intercepts

of its related quadratic function are the same.

Solving Quadratic Equations (10.4)

Standard Form of a Quadratic EquationA quadratic equation is an equation that can be written in the form , where a ≠ 0. This form is called thestandard form of a quadratic equation.

We can solve for some quadratic equations by graphing their related functions.


Sample Problem #1Solve the following quadratic equation by graphing the related function: x2 – 4 = 0.

Sample Problem #2Solve the following quadratic equation by graphing the related function: x2 = 0.

Sample Problem #3Solve the following quadratic equation by graphing the related function: x2 + 4 = 0.

We can also solve equations of the form x2 = a by finding the square roots.


Sample Problem Solve 2x2 – 98 = 0

We can use the zero-product property to solve quadratic equations when b ≠ 0 in the equation ax2 + bx + c = 0.

We will need to factor first then use the zero-product property to find the solutions.

Factoring to Solve Quadratic Equations (10.5)

Zero-Product PropertyFor every real number a and b, if ab = 0, then a = 0 or b = 0.

Sample Problem Solve (x + 5)(2x – 6) = 0

Sample Problem Solve x2 – 8x – 48 = 0 by factoring.


Sample Problem Solve 2x2 – 5x = 88.


Completing the Square (10.6)

The method of completing the square works for solving all kinds of quadratic equations. Finding the squares and factoring can only work for

solving some quadratic equations.Completing the square will turn every quadratic

equation into the form m2 = n. In completing the square we want to obtain a

trinomial that can then be factored. Once factored we can then solve for the quadratic

equation. Remember, in a perfect square trinomial with a=1, c must be the square of half of b (Sect.9.7) for y = ax2 + bx + c.

The process of completing the square is as follows for an equation in the form x2 + bx: Find half of the coefficient of x. Square the result of the first step. Add this result back into the original quadratic equation.


Sample ProblemFind the value of n such that x2 – 12x + n is a perfect square trinomial.

The simplest equations in which to complete the square have the form x2 + bx =c.


Sample ProblemSolve the equation x2 +9x = 136.

To solve for a quadratic equation in the form x2 + bx + c =0, first subtract the constant term c from each side of the equation.


Sample ProblemSolve the equation x2 – 20x + 32 = 0.

To solve for a quadratic equation in the form ax2 + bx – c = 0, we need to divide each side by a before completing the square.


Sample Problem Solve the equation 3x2 + 5x + 2 = 0.

Let’s complete the square for the general equation of a quadratic, ax2 + bx + c = 0.

Using the Quadratic Equation (10.7)

What we have just derived is an equation know as the quadratic formula.

The quadratic formula can also be used to solve any quadratic equation and has many uses outside of math. In using the quadratic formula to solve for real world

problems we have to determine if one or both answers would make sense in real-world situations.

Be sure to write a quadratic equation in standard form before using the quadratic formula.


Quadratic FormulaIf ax2 + bx + c = 0, and a ≠ 0, then

Sample ProblemSolve x2 + 6 = 5x


When the radicand in the quadratic formula is not a perfect square, use a calculator to approximate the solutions of an equation.

Using the Quadratic Equation (10.7) Sample Problem

Use the quadratic formula to solve the equation and then round the answers to the nearest hundredth.2x2 + 4x – 7 = 0

We can use the quadratic formula to solve all quadratic equations. However, sometimes another method may be easier.

Summary of the methods to solve a quadratic equation:


METHOD WHEN TO USE

Graphing Use if you have a graphing calculator handy.

Square Roots Use if the equation has no x term.

Factoring Use if you can factor the equation easily.

Completing the Square Use if the x2 term is 1, but you cannot factor the equation easily.

Quadratic Formula Use if the equation cannot be factored easily or at all.

Quadratic equations an have two, one, or no solutions.

We can determine how many solutions a quadratic has, before solving it, by using the discriminant. Discriminant: The expression under the radical sign in

the quadratic formula.

Using the Discriminant (10.8)

Property of the DiscriminantFor the quadratic equation ax2 + bx + c = 0 where a ≠ 0, you can use the value of the discriminant to determine the number of solutions.If b2 – 4ac > 0, there are two solutions (positive discriminant).If b2 – 4ac = 0, there is one solution.If b2 – 4ac < 0, there are no solutions (negative discriminant).

The relationship you see between the graphs and discriminant above is true for all cases (D is the discriminant in the graphs above).

In each case you are looking at the number of times the graph crosses the x-axis to determine the number of solutions.


Sample ProblemFind the number of solutions of 3x2 – 5x = 1.


Chapter 10

Quadratic Equations

and Functions

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Chapter 10. In this chapter we will examine quadratic graphs and their equations. We will solve quadratic equations by various techniques such as