Quadratic Equations and Functions Chapter 5 Quadratic Equations and Functions

In This Chapter You Will … Learn to use quadratic functions to model real-world data. Learn to graph and to solve quadratic equations. Learn to graph complex numbers and to use them in solving quadratic equations.

5.1 Modeling Data With Quadratic Functions 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. Solve using tables, graphs, and algebraic properties. Interpret the constants and coefficients in the context of the problem. What you’ll learn … To identify quadratic functions and graphs To model data with quadratic functions

Quadratic Functions and their Graphs A quadratic function is a function that can be written in the standard form, where a≠0. f(x) = ax2 + bx + c Quadratic term Linear term Constant term

Example 1 Classifying Functions Determine whether each function is linear or quadratic. Identify the quadratic, linear, and constant terms. y = (2x +3)(x – 4) f(x) = 3(x2-2x) – 3(x2 – 2)

The graph of a quadratic function is a parabola. The axis of symmetry is the line that divides a parabola into two parts that are mirror images.

The vertex of a parabola is the point at which the parabola intersects the axis of symmetry. The y-value of the vertex represents the maximum or minimum value of the function.

Example 2a Graph y = 2x2 – 8x + 8 Vertex ___________ Axis of Symmetry ______

Example 2b Graph y = -x2 – 4x + 2 Vertex ___________ Axis of Symmetry ______

Example 3a Finding a Quadratic Model Y 2 3 13 4 29 Find a quadratic function to model the values in the table. Substitute the values of x and y into y = ax2 + bx + c. The result is a system of three linear equations.

Example 3b Finding a Quadratic Model Y 1 2 -3 3 -10 Find a quadratic function to model the values in the table. Substitute the values of x and y into y = ax2 + bx + c. The result is a system of three linear equations.

Example 4 Real World Connection The table shows the height of a column of water as it drains from its container. Model the data with a quadratic function. Graph the data and the function. Use the model to estimate the water level at 35 seconds. Elapsed Time Water Level 0 s 120 mm 10 s 100 mm 20 s 83 mm 30 s 66 mm 40 s 50 mm 50 s 37 mm 60 s 28 mm Step 1 Enter data into L1 and L2. Use QuadReg. Step 2 Graph the data and the function. Step 3 Use the table to find f(35).

5.2 Properties of Parabolas 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. Solve using tables, graphs, and algebraic properties. Interpret the constants and coefficients in the context of the problem. What you’ll learn … To graph quadratic functions To find maximum and minimum values of quadratic functions

Graphing Parabolas The standard form of a quadratic function is y=ax2 + bx + c. When b=0, the function simplifies to y=ax2 + c. The graph of y=ax2 + c is a parabola with an axis of symmetry x =0, the y-axis. The vertex of the graph is the y-intercept (0,c).

Properties Graph of a Quadratic Function in Standard Form The graph of y=ax2 + bx + c is a parabola when a≠0. When a>0, the parabola opens up. When a<0, the parabola opens down. positive quadratic y = x2 negative quadratic y = –x2                                         

Properties Graph of a Quadratic Function in Standard Form The graph of y=ax2 + bx + c is a parabola when a≠0. The axis of symmetry is x= - b 2a

Properties Graph of a Quadratic Function in Standard Form The graph of y=ax2 + bx + c is a parabola when a≠0. The vertex is ( - , f(- ) ). b 2a b 2a

Properties Graph of a Quadratic Function in Standard Form The graph of y=ax2 + bx + c is a parabola when a≠0. The y intercept is (0,c).

Quadratic Graphs 2 y = x The graph of a quadratic function is a U-shaped curve called a parabola. .

Example 1 Graphing a Function of the Form y=ax2 + c Graph y= -½x2 + 2 Graph y= 2x2 - 4

Symmetry 2 y = x + 3 You can fold a parabola so that the two sides match evenly. This property is called symmetry. The fold or line that divides the parabola into two matching halves is called the axis of symmetry.

Vertex The highest or lowest point of a parabola is its vertex, which is on the axis of symmetry. 2 y = ½ x y = -4 x +3 2 Minimum Maximum

Determining Vertex and Axis of Symmetry Equation Max/Min Vertex Axis of Symmetry Y- Intercept(s) y = -x + 4x + 2 y = -1/3x - 2x-3 y = 2x + 8x -1 y = x - 2x - 3 2 2 2 2

5.3 Translating Parabolas What you’ll learn … To use the vertex form of a quadratic function 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. Solve using tables, graphs, and algebraic properties. Interpret the constants and coefficients in the context of the problem.

Investigation: Vertex Form Standard Form y = ax2 +bx + c Vertex Form y = a(x – h)2 + k h y = x2 -4x + 4 y = (x – 2)2 y = x2 +6x + 8 y = (x +3)2 - 1 y = -3x2 -12x - 8 y = -3(x +2)2 +4 y = 2x2 +12x +19 y = 2(x +3)2 +1 b 2a

In other words … To translate the graph of a quadratic function, you can use the vertex form of a quadratic function.

Properties The graph of y = a(x – h)2 + k is the graph of y = ax2 translated h units horizontally and k units vertically. When h is positive the graph shifts right; when h is negative the graph shifts left. When k is positive the graph shifts up; when the k is negative the graph shifts down. The vertex is (h,k) and the axis of symmetry is the line x=h.

Example 1a Using Vertex Form to Graph a Parabola 2 Graph y = - (x-2)2 +3 Graph the vertex. Draw the axis of symmetry. Find another point. When x=0. Sketch the curve.

Example 1b Using Vertex Form to Graph a Parabola Graph y = 2 (x+1)2 - 4 Graph the vertex. Draw the axis of symmetry. Find another point. When x=0. Sketch the curve.

Example 2a Writing the Equation of a Parabola Write the equation of the parabola. Use the vertex form. Substitute h=__ and k= ___. Substitute x=0 and y = 6. Solve for a.

Example 2b Writing the Equation of a Parabola Write the equation of the parabola. Use the vertex form. Substitute h=__ and k= ___. Substitute x=___ and y = ___. Solve for a.

Example 2c Writing the Equation of a Parabola Write the equation of a parabola that has vertex (-2, 1) and goes thru the point (1,28). Write the equation of a parabola that has vertex (-1, -4) and has a y intercept of 3.

Convert to Vertex Form y = 2x2 +10x +7 y = -3x2 +12x +5

Convert to Standard Form y = (x+3)2 - 1 y = -3(x -2 )2 +4

Example 3 Real World Connection The photo shows the Verrazano-Narrows Bridge in New York, which has the longest span of any suspension bridge in the US. A suspension cable of the bridge forma a curve that resembles a parabola. The curve can be modeled with the function y = 0.0001432(x-2130)2 where x and y are measured in feet. The origin of the function’s graph is at the base of one of the two towers that support the cable. How far apart are the towers? How high are they?

Start by drawing a diagram. The function, y = 0.0001432(x-2130)2 , is in vertex form. Since h =2130 and k =0, the vertex is (2130,0). The vertex is halfway between the towers, so the distance between the towers is 2(2130) ft = 4260 ft. To find the tower’s height, find y for x=0.

5.4 Factoring Quadratic Expressions What you’ll learn … To find common and binomial factors of quadratic expressions To factor special quadratic expressions 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

Investigation: Factoring Since 6 3 = 18, 6 and 3 up a factor pair for 18. Find the other factor pairs for 18, including negative integers. Find the sum of the integers in each factor pair for 18. Does 12 have a factor pair with a sum of -8? A sum of -9? Using all the factor pairs of 12, how many sums are possible? How many sums are possible for the factor pairs of -12? *

Factoring is rewriting an expression as the product of its factors. The greatest common factor (GCF) of an expression is the common factor with the greatest coefficient and the greatest exponent.

Example 1a Finding Common Factors 4x + 12 x - 8 GCF ________ 4b -2b -6b GCF ________ 3 2 3 2

Example 1b Finding Common Factors 3x - 12x +15x ( ) 6m - 12m - 24m ( ) 3 2 3 2 GCF GCF

Example 2 Factoring when ac>0 and b>0 Factor x2 +8x +7 Factor x2 +6x +8 Factor x2 +12x +32 Factor x2 +14x +40

Example 3 Factoring when ac>0 and b<0 Factor x2 -17x +72 Factor x2 -6x +8 Factor x2 -7x +12 Factor x2 -11x +24

Example 4 Factoring when ac<0 Factor x2 - x - 12 Factor x2 +3x - 10 Factor x2 -14x - 32 Factor x2 +4x - 5

Example 5 Factoring when a≠0 and ac>0 Factor 2x2 +11x + 12 Factor 3x2 - 16x +5 Factor 4x2 +7x + 3 Factor 2x2 - 7x + 6

Example 6 Factoring when a≠0 and ac<0 Factor 4x2 -4x - 15 Factor 2x2 +7x - 9 Factor 3x2 - 16x - 12 Factor 4x2 +5x - 6

Special Cases A perfect square trinomial is the product you obtain when you square a binomial. An expression of the form a2 - b2 is defined as the difference of two squares.

Factoring a Perfect Square Trinomial with a = 1 x - 8x + 16 n - 16n + 64 2 2

The Difference of Two Squares x - 121 ( ) ( ) 4x - 36 ( ) ( ) 2 2

5.5 Quadratic Equations What you’ll learn … To solve quadratic equations by factoring and by finding square roots To solve quadratic equations by graphing 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. Solve using tables, graphs, and algebraic properties. Interpret the constants and coefficients in the context of the problem.

The standard form of a quadratic equation is ax2 + bx + c = 0, where a ≠ 0. You can solve some quadratic equations in standard form by factoring the quadratic expression and then using the Zero-Product Property. Zero-Product Property If ab = 0, then a =0 or b=0. Example If (x +3) (x -7) = 0 then (x +3) = 0 or (x -7) = 0.

Zero Product Property ( x + 3)(x + 2) = 0 (x + 5)(2x – 3 ) = 0

Example 1a Solve by Factoring x – 8x – 48 = 0 x + x – 12 = 0 2 2

Example 1b Solve by Factoring 2x – 5x = 88 x - 12x = -36 2 2

Example 2 Solving by Finding Square Roots

Example 4 Solve by Graphing 2 2 x – 4 = 0 x = 0 2 x + 4 = 0 The number of x intercepts determines the number of solutions!!

Using the Calculator Solve:   1.  Set y= and graph with a standard window. 2.  Use the ZERO command to find the roots --  2nd TRACE (CALC), #2 zero 3.  Left bound?  Move the spider as close to the root (where the graph crosses the x-axis) as possible.  Hit the left arrow to move to the "left" of the root.  Hit ENTER.  A "marker" ► will be set to the left of the root.

4.  Right bound?  Move the spider as close to the root (where the graph crosses the x-axis) as possible.  Hit the right arrow to move to the "right" of the root.  Hit ENTER.  A "marker" ◄ will be set to the right of the root. 5.  Guess?  Just hit ENTER. 6.  Repeat the entire process to find the second root (which in this case happens to be x = 7).

Using a Graphing Calculator Solve Each Equation x2 + 6x + 4 = 0 3x2 + 5x - 12 = 8

5.6 Complex Numbers What you’ll learn … To identify and graph complex numbers To add, subtract, and multiply complex numbers 1.02 Define and compute with complex numbers.

When you learned to count, you used natural numbers 1,2,3, and so on When you learned to count, you used natural numbers 1,2,3, and so on. Your number system has grown to include other types of numbers. You have used real numbers, which include both rational numbers such as ½ and irrational numbers such as √2. Now your number system will expand to include numbers such as √-2.

The imaginary number i is defined as the number whose square is -1 The imaginary number i is defined as the number whose square is -1. So i2 = -1and i = √-1. An imaginary number is any number of the form a + bi where b≠0. Imaginary numbers and real numbers together make up the set of complex numbers. 6 + i - 4i 3 - 2i

Example 1 Simplifying Numbers Using i √-9 √-18 √-12 √-2

Example 2 Simplifying Imaginary Numbers √-9 + 6 √-18 + 7 a + bi

The diagram below shows the sets of numbers that are part of the complex number system and examples of each set.

You can use the complex number plane to represent a complex number geometrically. Locate the real part of the number on the horizontal axis and the imaginary part on the vertical axis. You graph 3 – 4i in the same way you would graph (3,-4) on the coordinate plane.

The absolute value of a complex number is its distance from the origin on the complex number plane. You can find the absolute value by using the Pythagorean Theorem. In general, a +bi = a2+b2

Example 3 Finding Absolute Values

Example 4 Additive Inverse of a Complex Number Find the additive inverse of -2 +5i. Find the additive inverse of 4 – 3i. Find the additive inverse of a + bi.

Example 5 Adding Complex Numbers Simplify (5 + 7i) + (-2 + 6i) Simplify (8 + 3i) - (2 + 4i) Simplify 7 - (3 + 2i)

Example 6 Multiplying Complex Numbers Find (5i) + (-4i) Find (2 + 3i) - (-3 + 5i) Find (6 – 5i) (4 – 3i)

Example 7 Finding Complex Solutions Solve 4x2 + 100 = 0 Solve 3x2 + 48 = 0 Solve -5x2 - 150 = 0

5.7 Completing the Square What you’ll learn … To solve equations by completing the square To rewrite functions by completing the square 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. Solve using tables, graphs, and algebraic properties. Interpret the constants and coefficients in the context of the problem.

Perfect Square Trinomials Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36

Creating a Perfect Square Trinomial In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____ Find the constant term by squaring half the coefficient of the linear term. (14/2)2 X2 + 14x + 49

Perfect Square Trinomials Create perfect square trinomials. x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___

Example 1 Solving a Perfect Square Trinomial Equation 2 Step 1: Factor the trinomial. Step 2: Find the Square Root of each side. Step 3: Solve for x. x +10x +25 = 36 2 (x+5) = 36 + x+5 = 6 x = 1 or x = -11

Example 2a Completing the Square x - 8x + __ 2 b 2 Find . Substitute -8 for b. Complete the square. -8 2 = 16 x - 8x + 16

Example 2b Completing the Square x + 7x + __ 2 b 2 Find . Substitute for b. Complete the square.

Example 3 Solving by Completing the Square Solve the following equation by completing the square: Step 1: Rewrite so all terms containing x are on one side.

Example 3 Continued Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.

Example 3 Continued Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation. Step 4: Take the square root of each side.

Example 3 Continued Step 5: Solve for x.

Solve each by Completing the Square x2 + 4x – 4 = 0 x2 – 2x – 1 = 0

Example 4 Finding Complex Solutions x2 - 8x + 36 = 0 x2 +6x = - 34

Example 5 Solving When a≠0 5x2 = 6x + 8 2x2 + x = 6

In lesson 5-3 you converted quadratic functions into vertex form by using x = - to find the x-coordinate of the parabola’s vertex. Then by substituting for x, you found the y coordinate of the vertex. Another way of rewriting a function is to complete the square. b 2a

Example 6a Rewriting in Vertex Form Complete the square. Add and subtract 3 on the right side. Factor the perfect square trinomial. Simplify. x2 + 6x + 2 2

Example 6b Rewriting in Vertex Form y = x2 - 10x - 2 y = x2 + 5x + 3

5.8 The Quadratic Formula What you’ll learn … To solve quadratic equations by using the quadratic formula To determine types of solutions by using the discriminant 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. Solve using tables, graphs, and algebraic properties. Interpret the constants and coefficients in the context of the problem.

The Quadratic Formula

Example 1a Using the Quadratic Formula x – 2x – 8 = 0 -b ± √ (b) – 4 (a) (c) 2(a) 2 - ( ) ± √ ( ) – 4 ( ) ( ) 2( ) 2 2

Example 1b Using the Quadratic Formula x – 4x – 117 = 0 -b ± √ (b) – 4 (a) (c) 2(a) 2 - ( ) ± √ ( ) – 4 ( ) ( ) 2( ) 2 2

Example 2a Finding Complex Solutions 2x = -6x - 7 -b ± √ (b) – 4 (a) (c) 2(a) 2 2

Example 2b Finding Complex Solutions -2x = 4x + 3 -b ± √ (b) – 4 (a) (c) 2(a) 2 2

Quadratic equations can have real or complex solutions Quadratic equations can have real or complex solutions. You can determine the type and number of solutions by finding the discriminant. -b + √ b2 – 4ac 2a x = ◄ the discriminant

Two imaginary solutions Value of the Discriminant Type and Number of Solutions for ax2 + bx + c Examples of Graphs of Related Functions y=ax2 + bx + c b2 – 4ac > 0 Two real solutions b2 – 4ac = 0 One real solution b2 – 4ac < 0 No real solution; Two imaginary solutions

Example 4 Using the Discriminant x +6x + 8 = 0 2 x +6x + 10 = 0 2

Methods for Solving Quadratics Discriminant Methods Positive square number Factoring, Graphing, Quadratic Formula, or Completing the Square Positive non-square number For approximate solutions: Graphing, Quadratic Formula, or Completing the Square For exact solutions: Quadratic Formula, or Completing the Square Zero Negative Quadratic Formula, or Completing the Square

In This Chapter You Should Have … Learned to use quadratic functions to model real-world data. Learned to graph and to solve quadratic equations. Learned to graph complex numbers and to use them in solving quadratic equations.