1. Quadratic Equations and Functions
Chapter 10

2. 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

3. Exploring Quadratic Graphs (10.1)
Standard Form of a Quadratic Function
A 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.
Examples: y = 5x2; y = x2 + 7; y = x2 – x – 3
The 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.

4. Exploring Quadratic Graphs (10.1)
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.

5. Exploring Quadratic Graphs (10.1)
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.
Exploring Quadratic Graphs (10.1)

6. Exploring Quadratic Graphs (10.1)
The value of a, the coefficient of the x2 term in the quadratic function, affects the width .
From the examples, we see that for 𝑚 < 𝑛 , 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.
Exploring Quadratic Graphs (10.1)

7. Exploring Quadratic Graphs (10.1)
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.
Exploring Quadratic Graphs (10.1)

8. Exploring Quadratic Graphs (10.1)
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.
Exploring Quadratic Graphs (10.1)

9. Exploring Quadratic Graphs (10.1)
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).
Exploring Quadratic Graphs (10.1)

10. Exploring Quadratic Graphs (10.1)
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.
Exploring Quadratic Graphs (10.1)

11. Quadratic Functions (10.2)
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)

12. Quadratic Functions (10.2)
Graph of a Quadratic Equation
The graph of y = ax2 + bx + c, where a ≠ 0, has
the line x = −𝑏 2𝑎 as its axis of symmetry.
the x-coordinate of the vertex is −𝑏 2𝑎 .
Quadratic Functions (10.2)
Examples:

13. Quadratic Functions (10.2)
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.
Quadratic Functions (10.2)
Sample Problem
Graph the function
y = -3x2 + 6x + 5

14. Quadratic Functions (10.2)
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 ≤or ≥.
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).
Quadratic Functions (10.2)

15. Finding and Estimating Square Roots (10.3)
The number a is a square root of b if a2 = b.
Every positive number ha two square roots.
A radical symbol (√) indicates a square root.
The expression 𝑎 means the positive, or principal square root.
The expression − 𝑎 means the negative square root.
The expression under the radical sign is known as the radicand.
Sample Problem
Simplify each expression. 64
− 100 ±
± 0

16. Finding and Estimating Square Roots (10.3)
Some square roots are rational numbers and some are irrational numbers.
Example: Rational roots
100 =10, ± = ±0.6, = 4 11
Example: Irrational roots
10 ≈ , ≈
Remember: In decimal form a rational number terminates or repeats, whereas an irrational number continues without repeating.
Finding and Estimating Square Roots (10.3)

17. Finding and Estimating Square Roots (10.3)
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.
Finding and Estimating Square Roots (10.3)
Sample Problem
Between what two consecutive integers is ?

18. Finding and Estimating Square Roots (10.3)
We can apply square roots to real world situations. Problem solving involving square roots.
Finding and Estimating Square Roots (10.3)
Sample Problem
The formula 𝑑= 𝑥 𝑥 2
gives the length d of each wire
for the tower at the right. Find
the length of the wire if x = 12 ft. d 2x x

19. Solving Quadratic Equations (10.4)
Standard Form of a Quadratic Equation
A quadratic equation is an equation that can be
written in the form 𝒂 𝒙 𝟐 +𝒃𝒙+𝒄=𝟎,
where a ≠ 0. This form is called the
standard form of a quadratic equation.
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.

20. Solving Quadratic Equations (10.4)
We can solve for some quadratic equations by graphing their related functions.
Solving Quadratic Equations (10.4)
Sample Problem #1
Solve the following quadratic
equation by graphing the
related function: x2 – 4 = 0.
Sample Problem #2
Solve the following quadratic
equation by graphing the
related function: x2 = 0.
Sample Problem #3
Solve the following quadratic
equation by graphing the
related function: x2 + 4 = 0.

21. Solving Quadratic Equations (10.4)
We can also solve equations of the form x2 = a by finding the square roots.
Solving Quadratic Equations (10.4)
Sample Problem
Solve 2x2 – 98 = 0

22. Factoring to Solve Quadratic Equations (10.5)
Zero-Product Property
For every real number a and b,
if ab = 0, then a = 0 or b = 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.
Sample Problem
Solve (x + 5)(2x – 6) = 0

23. Factoring to Solve Quadratic Equations (10.5)
Sample Problem Solve x2 – 8x – 48 = 0 by factoring.
Factoring to Solve Quadratic Equations (10.5)

24. Factoring to Solve Quadratic Equations (10.5)
Sample Problem Solve 2x2 – 5x = 88.
Factoring to Solve Quadratic Equations (10.5)

25. 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.
Completing the Square (10.6)

26. Completing the Square (10.6)
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.
Completing the Square (10.6)
Sample Problem
Find the value of n such that
x2 – 12x + n is a perfect square
trinomial.

27. Completing the Square (10.6)
The simplest equations in which to complete the square have the form x2 + bx =c.
Completing the Square (10.6)
Sample Problem
Solve the equation x2 +9x = 136.

28. Completing the Square (10.6)
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.
Completing the Square (10.6)
Sample Problem
Solve the equation x2 – 20x + 32 = 0.

29. Completing the Square (10.6)
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.
Completing the Square (10.6)
Sample Problem
Solve the equation 3x2 + 5x + 2 = 0.

30. Using the Quadratic Equation (10.7)
Let’s complete the square for the general equation of a quadratic, ax2 + bx + c = 0.
Using the Quadratic Equation (10.7)

31. 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.
Using the Quadratic Equation (10.7)
Quadratic Formula
If ax2 + bx + c = 0, and a ≠ 0, then
𝒙= −𝒃 ± 𝒃 𝟐 −𝟒𝒂𝒄 𝟐𝒂

32. Using the Quadratic Equation (10.7)
Sample Problem Solve x2 + 6 = 5x
Using the Quadratic Equation (10.7)

33. Using the Quadratic Equation (10.7)
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

34. Using the Quadratic Equation (10.7)
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:
Using the Quadratic Equation (10.7)
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.

35. Using the Discriminant (10.8)
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 Discriminant
For 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).

36. Using the Discriminant (10.8)
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.

37. Using the Discriminant (10.8)
Sample Problem Find the number of solutions of 3x2 – 5x = 1.
Using the Discriminant (10.8)

38. Quadratic Equations and Functions
Chapter 10
THE END