Algebra II Chapter 4: Quadratic Functions and Factoring Part 1

Transcription

1 Algebra II Chapter 4: Quadratic Functions and Factoring Part 1 Chapter 4 Lesson 1 Graph Quadratic Functions in Standard Form Vocabulary 1

2 Example 1: Graph a Function of the Form y = ax 2 Steps: 1. Make a table of values. You choose the domain. Then calculate the range. 2. Plot the points. 3. Draw a smooth curve through the points. Always draw the parent function!!! State the domain and range. Similarities: Differences: Domain: Range: Example 2: Graph a Function of the Form y = ax 2 + c Steps: 1. Make a table of values. You choose the domain. Then calculate the range. 2. Plot the points. 3. Draw a smooth curve through the points. Always draw the parent function!!! Compare the graph with the parent function. State the domain and range. Similarities: Differences: Domain: Range: 2

3 Example 3: Graph a Function of the Form y = ax 2 + bx + c Steps for graphing these quadratics: 1. Identify the coefficients and if it opens up or down. 2. Find the vertex. Use, then substitute to find y. 3. Draw the axis of symmetry. 4. Find two more points on the graph using a table of values. 5. Use symmetry to plot corresponding points. 6. Draw the curve through the points. Reminder: Gruver needs to talk about the solutions, roots, x intercepts, zeroes!!! Always draw the parent function!!! Compare the graph with the parent function. State the domain and range. Similarities: Differences: Domain: Range: Example 4: Find a Minimum or Maximum Value Hint: The minimum or maximum is always going to be a vertex. Find the vertex, and then decide if it is a maximum or minimum. 3

4 Example 5: Solve a Multi Step Problem Remind Gruver to give you the Graphing Sheet! Using a Graphing calculator to graph quadratic equations: 1. Make sure the quadratic is equal to y or simply Touch "y=" on your calculator. 3. Enter the quadratic equation. (You may add more than one if you need to graph more than one.) 4. Touch "GRAPH". 5. You may have to adjust your window by touching "WINDOW". You may also zoom in or out or locate a graph by touching "ZOOM". 6. You can approximate zeroes by finding in the table where the y values are 0. You can find a table of values by touching "2ND" then "TABLE". (TABLE is written in blue above the GRAPH button.) You can adjust the table settings by touching "2ND" then "TBLSET" (TBLSET is written in blue above the WINDOW button.) Using a graphing calculator to approximate zeroes: (This only works if the parabola crosses the x axis!!!! 1. Graph the equation. 2. Touch 2nd then CALC. 3. Scroll to 2:zero and touch ENTER. 4. Scroll to the left side of the graph where it crosses the x axis. Touch ENTER. 5. Scroll to the right side of the graph where it crosses the x axis. Touch ENTER. 6. Touch ENTER again. (This will give you one of the zeroes.) 4

5 Chapter 4 Lesson 3 Solve x 2 + bx + c = 0 by Factoring Vocabulary Example 1: Factor Trinomials of the Form x 2 + bx + c When the coefficient of the x 2 term is 1, find out which 2 factors of c will give you b. I like to use an x to help me. NOTE: Some trinomials CANNOT be factored!!! You could guess and check, or you could try the x method: Begin like this: product a*c b Problem: x 2 8x 9 sum 1. Idenfy a, b, and c. Fill in the x. Make sure there is not a GCF in the expression to begin with!!!! 2. Find what values the blank spaces in the x have to be to get a product of 9 and a sum of Write the new sets of parentheses. 5

6 Example 2: Factor with Special Patterns These are only to help you save time... A LOT OF TIME! You can factor them like we did before! Example 3: Standardized Test Practice 1. Factor first! 2. Put each set of parentheses equal to zero. 3. Solve. HINT: These answers are where the parabola crosses the x axis. They are also known as the "x intercepts", the "zeros", the "solutions" or the "roots". 6

7 Example 4: Use a Quadratic Equation as a Model Reminder: The equation must equal 0 if you are to factor it!!! Example 5: Find the Zeros of Quadratic Functions Don't forget to substitute 0 for y! Zero is the y value at an x intercept! 7

8 Chapter 4 Lesson 4 Solve ax 2 + bx + c = 0 by Factoring Vocabulary Example 1: Factor ax 2 + bx + c where c>0 Ask Gruver for the written directions! NOTE: Some trinomials CANNOT be factored!!! Remember, you can always guess and check or you can use x box. Factoring Using the X BOX Method!! :) Begin like this: product a*c First Term x Last Term x b Problem: 3x 2 + 5x 12 sum 1. Idenfy the coefficients and fill in the x. Make sure there is not a GCF in the expression to begin with!!!! 2. Find what values the blank spaces in the x have to be to get a product of 9 and a sum of 8. Put them in the box. 3. Factor out the Greatest Common Factor (GCF) terms in the up and going le direcons. These are the values that go into your parentheses. Hint: Be careful to figure out the signs!!!! If the x term is negave, pull out a negave. 8

9 Example 2: Factor ax 2 + bx + c where c<0 Example 3: Factor with Special Patterns These are only to help you save time! A LOT of time!! You can factor them like we did before! 9

10 Example 4: Factor Out Monomials First Always check to make sure there is not a GCF in the expression before you begin to factor!! If there is, it MUST be factored out, then you can proceed. Example 5: Solve Quadratic Equations Almost Same Process as Before: 1. Make sure the equation is equal to zero. 2. Factor first, if possible! Use all of your factoring knowledge now!!! :) 3. Put each set of parentheses equal to zero. 4. Solve. HINT: These answers are where the parabola crosses the x axis. They are also known as the "x intercepts", the "zeros", the "solutions" or the "roots". 10

11 Example 6: Use a Quadratic Equation as a Model Chapter 4 Lesson 5 Solve Quadratic Equations by Finding Square Roots Vocabulary 11

12 Simplified Radical Form This means simplifying a radical so that there are no more square roots left under the radical sign. It also means removing any radicals in the denominator of a fraction. To do this, follow these steps: 1. Complete the prime factorization for the number. 2. Locate the perfect squares. (Once factored, perfect squares will have two of the same numbers.) 3. Place the number that represent the perfect squares outside the radical sign. 4. Once all perfect squared numbers are outside the radical, multiply them back together. 5. Multiply the remaining numbers back together to determine what the number under the radical sign should be. 6. Make sure there are no radicals in the denominator of a fraction. If there are, follow these steps: a. Multiply the fraction by the radical over the radical or the conjugate over the conjugate. b. Simplify. Multiplying Numbers with Radicals 1. Multiply the numbers outside the radicals together. That's the new number outside the radical. 2. Multiply the numbers under the radicals together. That's the new number under the radical. 3. Simplify if needed. Dividing Radicals 1. Follow the directions for simplifying a fraction with a radical in the denominator! Example 1: Use Properties of Square Roots This is also known as simplifying! It MUST be done if it can be done!!! * 12

13 Example 2: Rationalize Denominators of Fractions You must always rationalize the denominator. That means to make sure there is no radical in the denominator of fractions! Some Helpful Ideas: Example 3: Solve a Quadratic Equation Use this method when you have no x term. Don't forget that you could have a positive or negative result! Solve for x using opposite operations. 13

14 Example 4: Standardized Test Practice Use this method when the quadratic is in vertex form. Don't forget that you could have a positive or negative result! Solve for x using opposite operations. Example 5: Model a Dropped Object with a Quadratic Function This is the formula to find the (h) height of a falling object after (t) seconds and being dropped from (s) initial height. 14