Ready to Go On? Skills Intervention 1-1. Exploring Transformations. 2 Holt McDougal Algebra 2. Name Date Class

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1 Lesson - Read to Go n? Skills Intervention Eploring Transformations Find these vocabular words in the lesson and the Multilingual Glossar. Vocabular transformation translation reflection stretch Translating and Reflecting Functions Use a table to perform each transformation of 5 f (). Use the same coordinate plane as the original function. A. translation down 3 units Graph the coordinates (, ). Then graph the coordinates (, 3). Complete the table What happens to the graph (, ) after it is translated down 3 units? B. reflection across ais Complete the table. 0 0 R Multipl each coordinate b. 5 Graph the coordinates (, ). Then graph the coordinates (, ) What happens to the graph (, ) after it is reflected across the ais? Copright b Holt, Rinehart and Winston. Holt McDougal Algebra

2 Lesson - Read to Go n? Problem Solving Intervention Eploring Transformations Gm Fees A local eercise gm charges different monthl fees depending on the length of the contract a person signs. The graph shows the various fees. Sketch a graph to represent each of the following situations and identif the transformation of the original graph that it represents. a. A coupon allows for monthl fees to be decreased b $5 per month. b. A rise in costs causes monthl fees to increase b 0%. Understand the Problem. What does the line on the graph show? Fee ($) Months of Contract. Upon what does the monthl fee depend? Make a Plan 3. If monthl fees decrease b $5, the -coordinate will remain the same and the -coordinate will decrease b. 4. If monthl fees increase b 0%, the -coordinate will remain the same and the -coordinate will increase b. Solve 5. List three points from the original graph. (, ); (8, ); (4, ) Write the new coordinates that result when monthl fees decrease b $5. (, ); (8, ); (4, ) Plot the new points on the graph. How is the graph translated? 6. Write the new coordinates that result when monthl fees increase b 0%. (, ); (8, ); (4, ) Plot the new points on the graph. How is the graph translated? Look Back 7. Look at the graphs. Do the translations match the change in the monthl fees? Eplain. Copright b Holt, Rinehart and Winston. 3 Holt McDougal Algebra

3 Lesson - Find this vocabular word in the lesson and the Multilingual Glossar. Identifing Transformations of Parent Functions Identif the parent function for g from its function rule. Then graph on our calculator and describe what transformation of the parent function it represents. A. g () 5 4 What is the power of in the function g () 5 4? Graph the function on our calculator. B. g () 5 3 What is the power of in the function g () 5 3? Graph the function on our calculator. C. g () 5 ( ) What is the power of in the function g () 5 ( )? Graph the function on our calculator. D. g () 5 3 Read to Go n? Skills Intervention Introduction to Parent Functions What is the power of in the function g () 5 3? Graph the function on our calculator. Vocabular parent function Copright b Holt, Rinehart and Winston. 4 Holt McDougal Algebra

4 Lesson - Read to Go n? Problem Solving Intervention Introduction to Parent Functions Parent functions can help ou sketch a curve to approimate those values not in a data table. The table lists the distance an object has fallen after a given number of seconds. Graph the relationship between distance and time and identif which parent function best describes this function. Then use the graph to estimate the distance the object will have fallen after 0 seconds. Understand the Problem Falling bject Time (s) Distance (ft) What information is shown in the table?. What are the input values? 3. What are the output values? Make a Plan 4. What variable should be plotted on the -ais of the graph? 5. What variable should be plotted on the -ais of the graph? Solve 6. List five points to plot on the graph based on the information in the table. (, 6); (, ); (, 44); (, ); (, ) 7. Graph the points ou listed in Eercise 6. Draw a smooth curve through them. 8. What is the shape of the graph? What is the parent function? 9. Estimate the distance traveled b the object after Distance (ft) Time (s) 0 seconds. Look Back 0. Etend the curve in the graph. Is it close to the estimate? Copright b Holt, Rinehart and Winston. 5 Holt McDougal Algebra

5 Section A Read to Go n? Quiz - Eploring Transformations The graph shows the cost of movie tickets at a particular theater. Sketch a graph to represent each situation and identif the transformation of the original graph that it represents.. The cost of a ticket increases b $3 for special movie premieres.. Senior citizens receive a discount of %. Cost ($) Number of Tickets - Introduction to Parent Functions Identif the parent function for from its equation. Then graph on our calculator and describe what transformation of the parent function it represents. 3. g() 5.5 Parent function: 4. g() 5 6 Parent function: 5. g() Parent function: 6. Graph the relationship between the number of cell phones sold and monthl income. Identif which parent function best describes the relationship. Then use the graph to estimate the monthl income when cell phones are sold. Cell Phone Sales Income Phones Sold Monthl Income ($) Monthl Income ($) Phones Sold Copright b Holt, Rinehart and Winston. 6 Holt McDougal Algebra

6 Section A Read to Go n? Enrichment Eploring Transformations Transform = f() through the series of changes described. Draw each transformation on the grids provided. A B. horizontal compression b a factor of. then a reflection across the -ais 3. then a translation down four units 4. then a vertical stretch b a factor of 3 Copright b Holt, Rinehart and Winston. 7 Holt McDougal Algebra

7 Lesson -3 Read to Go n? Skills Intervention Transforming Linear Functions Translating and Reflecting Linear Functions Let g ( ) be the indicated transformation of f ( ). Write the rule for g ( ). f ( ) 5 ; vertical translation 3 units up Does a vertical translation change the input values or the output values? What number is being added to each value? g( ) 5 f () Replace f() with the function given. g( ) 5 ( ) Simplif the final function. g( ) 5 Stretching and Compressing Linear Functions Let g ( ) be the indicated transformation of f ( ). Write the rule for g ( ). f ( ) 5 5; vertical compression b a factor of How does a vertical compression change the graph of a function? Does a vertical compression change the input values or the output values? Multipl f ( ) b the factor of the compression. g () 5? 5 Simplif the function. g () 5 Combining Transformations of Linear Functions Let g ( ) be the indicated transformation(s) of f ( ). Write the rule for g ( ). f ( ) 5 8; horizontal stretch b a factor of 4 followed b a horizontal translation to the right units What is the first transformation? Do the input values or the output values change? What is the function after the first transformation? h( ) 5 f b 5 What is the second transformation? How do ou translate a function horizontall to the right? h ( ) Perform the second transformation to find g(). g ( ) 5 h ( ) g () 5 g ( ) 5 Copright b Holt, Rinehart and Winston. 8 Holt McDougal Algebra

8 Lesson -4 Read to Go n? Skills Intervention Curve Fitting with Linear Models Find these vocabular words in the lesson and the Multilingual Glossar. Vocabular regression correlation line of best fit correlation coefficient Finding the Slope of a Line Find the slope of each line. Then write the equation that fits the data. A. 00 Does the line slant upward or downward? Predict if the slope is positive or negative. Select one point on the line and call it (, ). (0, ) Select another point on the line and call it (, ). (, ) Substitute these ordered pairs into the slope formula and solve for m. m m ( ) 5 5 Use the point-slope form. 5 ( ) Substitute the values for,, and m. 5 Distribute. Add to isolate. 5 Simplif. B. Does the line slant upward or downward? (0, 8) (, 0) Predict if the slope is positive or negative. Select one point on the line and call it (, ). (0, ) Select another point on the line and call it (, ). (, ) Substitute these ordered pairs into the slope formula and solve for m. m m ( ) 5 5 Use the point-slope form. 5 ( ) Substitute the values for,, and m. 5 Distribute. Add to isolate. 5 Simplif. Copright b Holt, Rinehart and Winston. 9 Holt McDougal Algebra

9 Lesson -4 Read to Go n? Problem Solving Intervention Curve Fitting with Linear Models A scatter plot is helpful in understanding the relationships between two variables. A particular compan has offices in the United States and in Ital. Job applicants must be able to read and speak both English and Italian. As part of the application process, prospective emploees must take a test on their knowledge of Italian. The personnel office compared the number of ears applicants studied Italian to their test scores. Make a scatter plot of the data, and then sketch a line of best fit and find its equation. Years of Stud Test Scores Understand the Problem. What two variables does the data describe?. What three things are ou asked to do? Make a Plan 3. Which variable should be plotted as the independent variable (input)? 4. Which variable should be plotted as the dependent variable (output)? Solve 5. How man data points can ou plot from the data? Plot these points on the grid provided. 6. Is the correlation positive (upward) or negative (downward)? 7. Draw a line that splits the data evenl above and below the line. What are two points on the line? (, ); (, ) 8. Use two points on the line, such as (, ) and (5, 88) to find the slope of the line. m Use the point (, ) and the slope from Eercise 8 to write the equation of the line in point slope form. 5 m( ) 5 ( ) Look Back 0. Tr related points in the equation from Eercise 9 to see if the answer is reasonable. For eample, substitute 3 for. Is the output value near the other points on the scatter plot? Test Scores Years of Stud Copright b Holt, Rinehart and Winston. 0 Holt McDougal Algebra

10 Section B Read to Go n? Quiz -3 Transforming Linear Functions Let g () be the indicated transformation(s) of f (). Write the rule for g (). 7. f () 5 3; vertical translation 3 units down 8. f () 5 4; vertical stretch b a factor of 4 9. f () 5 ; horizontal compression b a factor of followed b a horizontal 4 translation left 8 units 0. f () 5 4; horizontal translation 6 units right followed b a vertical compression b a factor of 3-4 Curve Fitting with Linear Models. A student has kept track of the relative humidit and the apparent room temperature. The results are shown in the table below. Relative Humidit (%) Apparent Room Temperature, (8F) Apparent Temperature ( F) Relative Humidit (%) a. Draw a scatter plot of the data using relative humidit as the independent variable. b. Use our graphing calculator to find the correlation coefficient and the equation of the line of best fit for the data. What does the slope of the best fit mean for this data? c. Use our equation to predict the apparent room temperature at a relative humidit of 45%. Copright b Holt, Rinehart and Winston. Holt McDougal Algebra

11 Section B Read to Go n? Enrichment Scatter Plots Match the correlation coefficient to the data it most likel describes. A B C D. r r r r Arrange the correlation coefficients in order from the weakest correlation to the strongest , 0.9, 0.5, , 0., 0.98, , 0.00, 0.0, , 0.909, 0.099, Identif each statement as true or false. 9. A scatter plot in which there is no relation between the data has a correlation coefficient close to Some scatter plots have a correlation coefficient that is greater than, which indicates an even stronger relation between the data values.. A correlation coefficient close to indicates a relation with a strong linear trend with a negative slope. Copright b Holt, Rinehart and Winston. Holt McDougal Algebra

12 SecTion - Read to Go n? Skills Intervention Eploring Transformations Find these vocabular words in the lesson and the Multilingual Glossar. Vocabular transformation translation reflection stretch Translating and Reflecting Functions Use a table to perform each transformation of 5 f (). Use the same coordinate plane as the original function. A. translation down 3 units Graph the coordinates (, ). Then graph the coordinates (, 3). Complete the table What happens to the graph (, ) after it is translated down 3 units? 5 SecTion - Read to Go n? Problem Solving Intervention Eploring Transformations Gm Fees A local eercise gm charges different monthl fees depending on the length of the contract a person signs. The graph shows the various fees. Sketch a graph to represent each of the following situations and identif the transformation of the original graph that it represents. a. A coupon allows for monthl fees to be decreased b $5 per month. b. A rise in costs causes monthl fees to increase b 0%. Understand the Problem. What does the line on the graph show?. Upon what does the monthl fee depend? Make a Plan 3. If monthl fees decrease b $5, the -coordinate will remain the same and the -coordinate will decrease b. 4. If monthl fees increase b 0%, the -coordinate will remain the same and Fee ($) Months of Contract B. reflection across ais Complete the table. 0 0 Graph the coordinates (, ). Then graph the coordinates (, ). R Multipl each coordinate b the -coordinate will increase b. Solve 5. List three points from the original graph. (, ); (8, ); (4, ) Write the new coordinates that result when monthl fees decrease b $5. (, ); (8, ); (4, ) Plot the new points on the graph. How is the graph translated? 6. Write the new coordinates that result when monthl fees increase b 0%. (, ); (8, ); (4, ) Plot the new points on the graph. How is the graph translated? What happens to the graph (, ) after it is reflected across the ais? 5 Look Back 7. Look at the graphs. Do the translations match the change in the monthl fees? Eplain. Copright b Holt, Rinehart and Winston. Holt McDougal Algebra Copright b Holt, Rinehart and Winston. 3 Holt McDougal Algebra SecTion - Find this vocabular word in the lesson and the Multilingual Glossar. Identifing Transformations of Parent Functions Identif the parent function for g from its function rule. Then graph on our calculator and describe what transformation of the parent function it represents. A. g () 5 4 What is the power of in the function g () 5 4? Graph the function on our calculator. B. g () 5 3 What is the power of in the function g () 5 3? Graph the function on our calculator. C. g () 5 ( ) What is the power of in the function g () 5 ( )? Graph the function on our calculator. D. g () 5 3 Read to Go n? Skills Intervention Introduction to Parent Functions What is the power of in the function g () 5 3? Graph the function on our calculator. Vocabular parent function Section 8B Quadratic Functions and Equations Graphing Circles b Completing the Square Completing the square can be used to graph circles. The general equation for a circle with its center at the origin is 5 r, where r is the radius of the circle. The general equation of a circle with its center translated from the origin is ( h ) ( k ) 5 r. An equation representing a circle can be transformed into the sum of two squares. Eample: ( 4 ) ( 6 ) 5 49 ( 4 49) ( 6 9) ( 7 ) ( 3 ) 5 9 ( 7 ) ( 3 ) The center of the circle is (7, 3) and the radius is 3. 6 The circle is shown at the right. Complete the square on the following equations. Identif the center and radius of the circle and then graph Center: Radius: Read To Go n? Enrichment Center: Radius: Copright b Holt, Rinehart and Winston. 4 Holt McDougal Algebra Center: Radius: Center: Radius: Copright b Holt McDougal. 69 Holt McDougal Algebra Copright b Holt McDougal. 96 Holt McDougal Algebra

13 SecTion A Read to Go n? Quiz - Eploring Transformations Eploring Transformations The graph shows the cost of movie tickets at a particular theater. Sketch a graph to represent each situation and identif the Transform = f() through the series of changes described. Draw each transformation on the grids provided. transformation of the original graph that it represents.. The cost of a ticket increases b $3 for special movie premieres. 0 0 A B Senior citizens receive a discount of %. Number of Tickets Cost ($) Section A Read to Go n? Enrichment - Introduction to Parent Functions Identif the parent function for from its equation. Then graph on our calculator and describe what transformation of the parent function it represents. 3. g() 5.5 Parent function: 4. g() 5 6. horizontal compression b a factor of. then a reflection across the -ais Parent function: 5. g() Parent function: 6. Graph the relationship between the number of cell phones sold and monthl income. Identif which parent function best describes the relationship. Then use the graph to estimate the monthl income when cell phones are sold. Cell Phone Sales Income Phones Sold Monthl Income ($) Monthl Income ($) Phones Sold 3. then a translation down four units 4. then a vertical stretch b a factor of 3 Copright b Holt, Rinehart and Winston. 6 Holt McDougal Algebra Copright b Holt, Rinehart and Winston. 7 Holt McDougal Algebra SecTIn -3 Read to Go n? Skills Intervention Transforming Linear Functions Translating and Reflecting Linear Functions Let g () be the indicated transformation of f (). Write the rule for g (). f () 5 ; vertical translation 3 units up Does a vertical translation change the input values or the output values? What number is being added to each value? g( ) 5 f () Replace f() with the function given. g( ) 5 ( ) Simplif the final function. g( ) 5 Stretching and Compressing Linear Functions Let g () be the indicated transformation of f (). Write the rule for g (). f () 5 5; vertical compression b a factor of How does a vertical compression change the graph of a function? Does a vertical compression change the input values or the output values? Multipl f ( ) b the factor of the compression. g () 5 Simplif the function. g () 5 Combining Transformations of Linear Functions Let g () be the indicated transformation(s) of f (). Write the rule for g (). f () 5 8; horizontal stretch b a factor of 4 followed b a horizontal translation to the right units What is the first transformation? Do the input values or the output values change?? 5 What is the function after the first transformation? h( ) 5 f b 5 What is the second transformation? How do ou translate a function horizontall to the right? Perform the second transformation to find g(). h ( ) g ( ) 5 h ( ) g ( ) 5 g ( ) 5 Copright b Holt, Rinehart and Winston. 8 Holt McDougal Algebra SectioN -4 Read to Go n? Skills Intervention Curve Fitting with Linear Models Find these vocabular words in the lesson and the Multilingual Glossar. Vocabular regression correlation line of best fit correlation coefficient Finding the Slope of a Line Find the slope of each line. Then write the equation that fits the data. A. 00 Does the line slant upward or downward? Predict if the slope is positive or negative. Select one point on the line and call it (, ). (0, ) Select another point on the line and call it (, ). (, ) Substitute these ordered pairs into the slope formula and solve for m. m m ( ) 5 5 Use the point-slope form. 5 ( ) Substitute the values for,, and m. 5 Distribute. Add to isolate. 5 Simplif. B. Does the line slant upward or downward? (0, 8) (, 0) Predict if the slope is positive or negative. Select one point on the line and call it (, ). (0, ) Select another point on the line and call it (, ). (, ) Substitute these ordered pairs into the slope formula and solve for m. m m ( ) 5 5 Use the point-slope form. 5 ( ) Substitute the values for,, and m. 5 Distribute. Add to isolate. 5 Simplif. Copright b Holt, Rinehart and Winston. 9 Holt McDougal Algebra Copright b Holt McDougal. 97 Holt McDougal Algebra

14 SecTion -4 Read to Go n? Problem Solving Intervention Curve Fitting with Linear Models A scatter plot is helpful in understanding the relationships between two variables. A particular compan has offices in the United States and in Ital. Job applicants must be able to read and speak both English and Italian. As part of the application process, prospective emploees must take a test on their knowledge of Italian. The personnel office compared the number of ears applicants studied Italian to their test scores. Make a scatter plot of the data, and then sketch a line of best fit and find its equation. Years of Stud Test Scores Understand the Problem. What two variables does the data describe?. What three things are ou asked to do? Make a Plan 3. Which variable should be plotted as the independent variable (input)? 4. Which variable should be plotted as the dependent variable (output)? Solve 5. How man data points can ou plot from the data? Plot these points on the grid provided. 6. Is the correlation positive (upward) or negative (downward)? 7. Draw a line that splits the data evenl above and below the line. What are two points on the line? (, ); (, ) 8. Use two points on the line, such as (, ) and (5, 88) to find the slope of the line. m Use the point (, ) and the slope from Eercise 8 to write the equation of the line in point slope form. 5 m( ) 5 ( ) Look Back 0. Tr related points in the equation from Eercise 9 to see if the answer is reasonable. For eample, substitute 3 for. Is the output value near the other points on the scatter plot? Test Scores 4 6 Years of Stud Copright b Holt, Rinehart and Winston. 0 Holt McDougal Algebra SEction B -3 Transforming Linear Functions Let g () be the indicated transformation(s) of f (). Write the rule for g (). 7. f () 5 3; vertical translation 3 units down 8. f () 5 4; vertical stretch b a factor of 4 9. f () 5 ; horizontal compression b a factor of followed b a horizontal 4 translation left 8 units 0. f () 5 4; horizontal translation 6 units right followed b a vertical compression -4 b a factor of 3 Curve Fitting with Linear Models. A student has kept track of the relative humidit and the apparent room temperature. The results are shown in the table below. Relative Humidit (%) Read to Go n? Quiz Apparent Room Temperature, (8F) a. Draw a scatter plot of the data using relative humidit as the independent variable. b. Use our graphing calculator to find the correlation coefficient and the equation of the line of best fit for the data. What does the slope of the best fit mean for this data? c. Use our equation to predict the apparent room temperature at a relative humidit of 45%. Apparent Temperature ( F) Relative Humidit (%) Copright b Holt, Rinehart and Winston. Holt McDougal Algebra SeCTion B Read to Go n? Enrichment Scatter Plots Match the correlation coefficient to the data it most likel describes. A B C D Section - Read To Go n? Skills Intervention Using Transformations to Graph Quadratic Functions Find these vocabular words in the lesson and the Multilingual Glossar. Vocabular quadratic function parabola verte of a parabola verte form. r r r r Arrange the correlation coefficients in order from the weakest correlation to the strongest , 0.9, 0.5, , 0., 0.98, , 0.00, 0.0, , 0.909, 0.099, Identif each statement as true or false. 9. A scatter plot in which there is no relation between the data has a correlation coefficient close to Some scatter plots have a correlation coefficient that is greater than, which indicates an even stronger relation between the data values.. A correlation coefficient close to indicates a relation with a strong linear trend with a negative slope. Translating Quadratic Functions Using the graph of f ( ) as a guide, describe the transformations, and then graph the function. g () 5 ( 3 ) f ( h) 5 ( h ) represents the general form for a horizontal shift. If h, 0 the graph moves left and if h. 0 the graph moves. f () k 5 k represents the general form for a vertical shift. If k is negative the graph is shifted down and if k is positive the graph is shifted. g () 5 ( 3 ) 5 ( (3) ) Rewrite to identif h and k. Because h 5, the graph is translated 3 units left and since k 5, the graph is translated Complete the table of values and graph. f () 5 ( 3 ) (, f ()) 5 f (5) 5 (5 3 ) 5 3 (5, 3) 4 f (4) 5 (4 3 ) 5 (4, ) 3 f (3) 5 (3 3 ) 5 (3, ) f () 5 ( 3 ) 5 (, ) f () 5 ( 3 ) 5 (, ) unit down. Writing Transformed Quadratic Functions Use the description to write the quadratic function in verte form: f ( ) is verticall stretched b a factor of 3 and translated 4 units left. The form of a quadratic function is f () 5 a( h ) k. The a indicates a across the -ais and/or a vertical or compression. The h represents a translation and indicates a vertical translation. Vertical stretch b 3: means 5 3. Translated 4 units left means h 5. Substitute to write the transformed function. g () 5 a( h ) k g () 5 ( ) 0 g () 5 ( ) Copright b Holt, Rinehart and Winston. Holt McDougal Algebra Copright b Holt, Rinehart and Winston. 3 Holt McDougal Algebra Copright b Holt McDougal. 98 Holt McDougal Algebra