 # Chapter 8 Review Quadratic Functions.

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Graphing Quadratic Equations in Two Variables
§ 8.3 Graphing Quadratic Equations in Two Variables

We spent a lot of time graphing linear equations in chapter 3. The graph of a quadratic equation is a parabola. The highest point or lowest point on the parabola is the vertex. Axis of symmetry is the line that runs through the vertex and through the middle of the parabola.

Example x y Graph y = 2x2 – 4. (–2, 4) (2, 4) x y 2 4 1 –2 (–1, – 2) (1, –2) –4 –1 –2 (0, –4) –2 4

Intercepts of the Parabola
Although we can simply plot points, it is helpful to know some information about the parabola we will be graphing prior to finding individual points. To find x-intercepts of the parabola, let y = 0 and solve for x. To find y-intercepts of the parabola, let x = 0 and solve for y.

Characteristics of the Parabola
If the quadratic equation is written in standard form, y = ax2 + bx + c, 1) the parabola opens up when a > 0 and opens down when a < 0. 2) the x-coordinate of the vertex is To find the corresponding y-coordinate, you substitute the x-coordinate into the equation and evaluate for y.

Example Graph y = –2x2 + 4x + 5. x y (1, 7) Since a = –2 and b = 4, the graph opens down and the x-coordinate of the vertex is (0, 5) (2, 5) x y 3 –1 (–1, –1) (3, –1) 2 5 1 7 5 –1 –1

The Graph of a Quadratic Function
The vertical line which passes through the vertex is called the Axis of Symmetry or the Axis Recall that the equation of a vertical line is x =c For some constant c x coordinate of the vertex The y coordinate of the vertex is The Axis of Symmetry is the x =

Opens up when a>o opens down a < 0 Vertex axis of symmetry

Identify the Vertex and Axis of Symmetry of a Quadratic Function
Vertex =(x, y). thus Vertex = Axis of Symmetry: the line x = Vertex is minimum point if parabola opens up Vertex is maximum point if parabola opens down

Identify the Vertex and Axis of Symmetry
Vertex x = y = Vertex = (-1, -3) Axis of Symmetry is x = = -1 = -3 = -1

8.5 Quadratic Solutions The number of real solutions is at most two.
No solutions One solution Two solutions

Identifying Solutions
Example f(x) = x2 - 4 Solutions are -2 and 2.

Identifying Solutions
Now you try this problem. f(x) = 2x - x2 Solutions are 0 and 2.

The graph of a quadratic equation is a parabola. The roots or zeros are the x-intercepts. The vertex is the maximum or minimum point. All parabolas have an axis of symmetry.

One method of graphing uses a table with arbitrary x-values. Graph y = x2 - 4x Roots 0 and 4 , Vertex (2, -4) , Axis of Symmetry x = 2 x y 1 -3 2 -4 3 4

Try this problem y = x2 - 2x - 8. Roots Vertex Axis of Symmetry x y -2 -1 1 3 4

A quadratic equation is written in the Standard Form,
8.6 – Solving Quadratic Equations by Factoring A quadratic equation is written in the Standard Form, where a, b, and c are real numbers and Examples: (standard form)

If a and b are real numbers and if , then or .
8.6 – Solving Quadratic Equations by Factoring Zero Factor Property: If a and b are real numbers and if , then or Examples:

If a and b are real numbers and if , then or .
8.6 – Solving Quadratic Equations by Factoring Zero Factor Property: If a and b are real numbers and if , then or Examples:

Solving Quadratic Equations: 1) Write the equation in standard form.
8.6 – Solving Quadratic Equations by Factoring Solving Quadratic Equations: 1) Write the equation in standard form. 2) Factor the equation completely. 3) Set each factor equal to 0. 4) Solve each equation. 5) Check the solutions (in original equation).

8.6 – Solving Quadratic Equations by Factoring

8.6 – Solving Quadratic Equations by Factoring
If the Zero Factor Property is not used, then the solutions will be incorrect

8.6 – Solving Quadratic Equations by Factoring

8.6 – Solving Quadratic Equations by Factoring

8.6 – Solving Quadratic Equations by Factoring

8.6 – Solving Quadratic Equations by Factoring

8.6 – Solving Quadratic Equations by Factoring

8.6 – Quadratic Equations and Problem Solving
A cliff diver is 64 feet above the surface of the water. The formula for calculating the height (h) of the diver after t seconds is: How long does it take for the diver to hit the surface of the water? seconds

8.6 – Quadratic Equations and Problem Solving
The square of a number minus twice the number is 63. Find the number. x is the number.

8.6 – Quadratic Equations and Problem Solving
The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. What are the length and the width of the garden? The width is w. The length is w+5. feet feet

8.6 – Quadratic Equations and Problem Solving
Find two consecutive odd numbers whose product is 23 more than their sum? Consecutive odd numbers:

8.6 – Quadratic Equations and Problem Solving
The length of one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hypotenuse is 13 meters. What are the lengths of the legs? meters meters

Solving Quadratic Equations by the Square Root Property
§ 8.7 Solving Quadratic Equations by the Square Root Property

Square Root Property We previously have used factoring to solve quadratic equations. This chapter will introduce additional methods for solving quadratic equations. Square Root Property If b is a real number and a2 = b, then

Square Root Property Example Solve x2 = 49 Solve 2x2 = 4 x2 = 2
Solve (y – 3)2 = 4 y = 3  2 y = 1 or 5

Square Root Property Example Solve x2 + 4 = 0 x2 = 4
There is no real solution because the square root of 4 is not a real number.

Square Root Property Example Solve (x + 2)2 = 25 x = 2 ± 5
x = 2 + 5 or x = 2 – 5 x = 3 or x = 7

Square Root Property Example Solve (3x – 17)2 = 28 3x – 17 =

Solving Quadratic Equations by Completing the Square
§ 8.8 Solving Quadratic Equations by Completing the Square

Completing the Square In all four of the previous examples, the constant in the square on the right side, is half the coefficient of the x term on the left. Also, the constant on the left is the square of the constant on the right. So, to find the constant term of a perfect square trinomial, we need to take the square of half the coefficient of the x term in the trinomial (as long as the coefficient of the x2 term is 1, as in our previous examples).

Completing the Square Example x2 – 10x x2 + 16x x2 – 7x
What constant term should be added to the following expressions to create a perfect square trinomial? x2 – 10x add 52 = 25 x2 + 16x add 82 = 64 x2 – 7x add

Completing the Square Example
We now look at a method for solving quadratics that involves a technique called completing the square. It involves creating a trinomial that is a perfect square, setting the factored trinomial equal to a constant, then using the square root property from the previous section.

Completing the Square Solving a Quadratic Equation by Completing a Square If the coefficient of x2 is NOT 1, divide both sides of the equation by the coefficient. Isolate all variable terms on one side of the equation. Complete the square (half the coefficient of the x term squared, added to both sides of the equation). Factor the resulting trinomial. Use the square root property.

Solving Equations Example Solve by completing the square. y2 + 6y = 8
y = 4 or 2

Solving Equations Example (y + ½)2 = Solve by completing the square.
y2 + y – 7 = 0 y2 + y = 7 y2 + y + ¼ = 7 + ¼ (y + ½)2 =

Solving Equations Example Solve by completing the square.
2x2 + 14x – 1 = 0 2x2 + 14x = 1 x2 + 7x = ½ x2 + 7x = ½ = (x + )2 =

The Quadratic Formula Another technique for solving quadratic equations is to use the quadratic formula. The formula is derived from completing the square of a general quadratic equation.

The Quadratic Formula A quadratic equation written in standard form, ax2 + bx + c = 0, has the solutions.

Solve 11n2 – 9n = 1 by the quadratic formula. 11n2 – 9n – 1 = 0, so a = 11, b = -9, c = -1

Solve x2 + x – = 0 by the quadratic formula. x2 + 8x – 20 = (multiply both sides by 8) a = 1, b = 8, c = 20

Solve x(x + 6) = 30 by the quadratic formula. x2 + 6x + 30 = 0 a = 1, b = 6, c = 30 So there is no real solution.

The Discriminant The expression under the radical sign in the formula (b2 – 4ac) is called the discriminant. The discriminant will take on a value that is positive, 0, or negative. The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively.

The Discriminant Example
Use the discriminant to determine the number and type of solutions for the following equation. 5 – 4x + 12x2 = 0 a = 12, b = –4, and c = 5 b2 – 4ac = (–4)2 – 4(12)(5) = 16 – 240 = –224 There are no real solutions.

Steps in Solving Quadratic Equations If the equation is in the form (ax+b)2 = c, use the square root property to solve. If not solved in step 1, write the equation in standard form. Try to solve by factoring. If you haven’t solved it yet, use the quadratic formula.

Solving Equations Example Solve 12x = 4x2 + 4. 0 = 4x2 – 12x + 4
Let a = 1, b = -3, c = 1

Solving Equations Example Solve the following quadratic equation.

The Quadratic Formula Solve for x by completing the square.

Yes, you can remember this formula
Pop goes the Weasel Gilligan’s Island This one I can’t explain

How does it work Equation:

How does it work Equation:

The Discriminant The number in the square root of the quadratic formula.

The Discriminant The Discriminant can be negative, positive or zero
If the Discriminant is positive, there are 2 real answers. If the square root is not a prefect square ( for example ), then there will be 2 irrational roots ( for example ).

The Discriminant The Discriminant can be negative, positive or zero
If the Discriminant is positive, there are 2 real answers. If the Discriminant is zero, there is 1 real answer. If the Discriminant is negative, there are 2 complex answers. complex answer have i.

Describe the roots Tell me the Discriminant and the type of roots

Describe the roots 0, One rational root
Tell me the Discriminant and the type of roots 0, One rational root

Describe the roots 0, One rational root
Tell me the Discriminant and the type of roots 0, One rational root

Describe the roots 0, One rational root -11, Two complex roots
Tell me the Discriminant and the type of roots 0, One rational root -11, Two complex roots

Describe the roots 0, One rational root -11, Two complex roots
Tell me the Discriminant and the type of roots 0, One rational root -11, Two complex roots

Describe the roots 0, One rational root -11, Two complex roots
Tell me the Discriminant and the type of roots 0, One rational root -11, Two complex roots 80, Two irrational roots