3.4 and 4.3 Explain Graphing and Writing Linear Equations in Standard Form - Notes

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1 3.4 and 4.3 Explain Graphing and Writing Linear Equations in Standard Form - Notes Essential Question: How can you describe the graph of the equation Ax + By = C? How can you write the equation of a line in standard form? Main Ideas/ Questions What You Will Learn Notes/Examples To find x-intercepts, y-intercepts, and zeros of functions. To write and graph linear equations in standard form. To use linear equations to solve real-life problems. Vocabulary x-intercept: (, ) x-coordinate of a point where the graph crosses the x-axis. It occurs when y = 0. Also called a zero of a function. The zero is the x-value when y = 0 or (f(x) = 0). y-intercept: (, ) y-coordinate of a point where the graph crosses the y-axis. It occurs when x = 0. Practice: Identify the x-intercept, y-intercept, and zero of the following equations x-intercept: x-intercept: x-intercept: y-intercept: y-intercept: y-intercept: zero: zero: zero: Practice: Find the zero ( - intercept) of the given function. (Plug in for.) 4. 2x + 3y = x y = f(x) = 2x An artist rents a booth at an art show for $300. The function f(x) = 50x 300 represents the artist s profit, where x is the number of paintings the artist sells. a. Find the zero of the function. b. Explain what the zero means in this situation.

2 3.4 and 4.3 Explain Graphing and Writing Linear Equations in Standard Form - Notes Writing Equations in Standard Form - Ax + By = C **Equivalent to Slope-intercept Form and Point-Slope Form** Practice: Write the following equations in standard form. 8. y ( 2) = 3(x 2) 9. y 3 = 2 3 (x + 3) 10. y = 1 4 x students are going on a trip to the library. They will travel in small and large vans. A small van holds 8 people, and a large van holds 12 people. Write an equation in standard form that represents the number of small and large vans that your class can fill. Let s = Let l = Equation: 12. A banquet room has large and small tables. A small table seats 4 people and a large table seats 6 people. If 60 people plan to attend the banquet, write an equation in standard form that models the number of small and large tables needed for the banquet. Let s = Let l = Equation: 13. A dog kennel charges $25 per night to board your dog. The kennel also sells dog treats for $5 each. If you have $100 to spend, write an equation in standard form that models the possible combination of nights at the kennel and treats that you can buy. Let x = Let y = Equation: 14. The school band is selling sweatshirts and baseball caps to raise $9000 to attend a band competition. Sweatshirts cost $25 each and baseball caps cost $10 each. Let x = Let y = Equation: If 258 sweatshirts are sold, how many baseball caps are sold? Graph the equation. Find two more possible solutions in the context of the problem.

3 4.2 Explain Writing Equations in Point-Slope Form - Notes Essential Question: How can you write an equation of a line when you are given the slope and a point on the line? Main Ideas/ Questions What You Will Learn Notes/Examples To write an equation of a line given its slope and a point on the line. To write an equation of a line given two points on the line. To use linear equations to solve real-life problems Slope-intercept Form Point-slope Form Standard Form **Linear Equations can be written in 3 different forms, but all are EQUIVALENT (same line)** Point-slope form: y y 1 = m(x x 1 ) Where: (x1, y1) represents one point on the line m represents the slope of the line (x, y) represents all the points on the line Point- Slope Form ***Use the point-slope form when you are given Practice: Identify the slope and a point that the line passes through then graph the line. 1. y + 2 = 3(x 2) 2. y 1 = 2 (x + 1 ) 2 2 m = m = Point: Point: If given a POINT and a SLOPE Step 1: Identify the slope, m. Writing Equations Given a Point and a Slope Step 2: Label the point x 1 and y 1 Step 3: Substitute the slope, m, and point (x 1, y 1) into the formula. Practice: Write an equation in point-slope form of the line that passes through the point and has the given slope. 3. (3, 1); m = 2 4. (4, 0); slope = 2 3 y = (x ) y = (x )

4 4.2 Explain Writing Equations in Point-Slope Form - Notes Writing Equations Given Two Points If given TWO Points. Step 1: Find the slope Step 2: Use the Point-slope using: y 2 y 1 form: y y x 2 x 1 = m(x x 1 ) 1 Practice: Write an equation in point-slope form of the points, graph or values below. 5. (-4, 6) and (-2, 5) f(4) = 2 & f(12) = Find the y-intercept for #5 (write in slope-intercept form). 9. What conclusion can be drawn about the point-slope equation and slope-intercept equation for #5? 10. The student council is ordering customized foam hands to promote school spirit. The table shows the cost of ordering different numbers of foam hands. a. Write a linear model in point-slope form that represents the cost as a function of the number of foam hands. 11. Craig is driving at a constant speed of 60 miles per hour. After driving 3 hours, his odometer reads 265 miles. a. What information is given? Slope/y-intercept or Point/Slope m = b = Point: Slope: b. Write a linear function that represents the miles driven after x hours in point-slope form. c. Write the above equation in slope-intercept form. d. What does the odometer read after 7 hours of continuous driving?

5 4.4 Explain Writing Equations of Parallel and Perpendicular Lines - Notes Essential Question: How can you recognize lines that are parallel or perpendicular? Main Ideas/ Questions Notes/Examples What You Will Learn To identify and write equations of parallel lines. To identify and write equations of perpendicular lines. Parallel Lines: Lines in the same plane that never intersect. Parallel and Perpendicular Lines Perpendicular Lines: Two lines in the same plane that intersect to form right angles. Same slope (m), different y-intercepts (b) The product of their slopes (m) is -1 (Opposite reciprocal) Examples of parallel lines: Examples of opposite reciprocals: 3. and Multiply: and Multiply: Identifying Parallel and Perpendicular Lines: Find the slope for the following lines and then determine if the lines are parallel, perpendicular, or neither. Explain your answer. 5. m = 6. m = m = m = The lines are because The lines are because 7. Line 1: (2, 0), (-2, 2) m = 8. m = Line 2: (1, -2), (4, 4) m = m = The lines are because The lines are because

6 4.4 Explain Writing Equations of Parallel and Perpendicular Lines - Notes Writing Equations of Parallel Lines: Write the equation of a line in slope-intercept form that passes through the given point and is parallel to the given line. 7. ; 8. ; Writing Equations of Perpendicular Lines: Write the equation of a line in slope-intercept form that passes through the given point and is perpendicular to the given line. 9. ; 10. ; What is the slope of a line parallel to the x-axis? Perpendicular to the x-axis? What is the slope of a line parallel to the y-axis? Perpendicular to the y-axis? 11. Write the equation of a line parallel to that passes through 12. Write the equation of a line perpendicular to that passes through. ***Vertical lines are to horizontal lines.*** 13. Application: The position of a helicopter search and rescue crew is shown in the graph. The shortest flight path to the shoreline is one that is perpendicular to the shoreline. Write an equation that represents this path.

7 4.5 and 4.6 Explain Scatter Plots and Lines of Fit - Classwork 1. The table shows the weekly sales of a DVD and the number of weeks since its release. a. Make a scatter plot of the data. b. What type of correlation? Positive/Negative/No Correlation c. Draw a trend line. d. Use a calculator to find the equation of the line of best fit. (Round to the hundredths place if needed.) e. Interpret the slope and y-intercept of the line of best fit. Slope: y-intercept: f. Find the correlation coefficient, r: g. Determine the strength of the correlation coefficient. 2. The scatter plot shows the days x of practice and the numbers y of free throws made during practice. a. What type of correlation? Positive/Negative/No Correlation b. Draw a trend line. c. Use a calculator to find the equation of the line of best fit. (Round to the hundredths place if needed.) c. Find the correlation coefficient, r: d. Determine the strength of the correlation coefficient. f. Predict the number of free throws made after 10 days of practice.

8 4.5 and 4.6 Explain Scatter Plots and Lines of Fit - Classwork 3. The table shows the durations x (in minutes) of several eruptions of the geyser Old Faithful and the times y (in minutes) until the next eruption. a. What type of correlation? Positive/Negative/No Correlation b. Use a calculator to find the equation of the line of best fit. (Round to the hundredths place if needed.) c. Find the correlation coefficient, r: d. Determine the strength of the correlation coefficient. e. Approximate the duration before a time of 77 minutes. f. Predict the time after an eruption lasting 5.0 minutes. Practice: Tell whether a correlation is likely in the situation. If so, tell whether there is a causal relationship. Explain your reasoning. 4. time spent exercising and the number of calories burned Correlation: Causation: YES/NO 5. the number of banks and the population of a city Correlation: Causation: YES/NO 6. time spent playing video games and grade point average Correlation: Causation: YES/NO 7. Eating organic food and getting a good score on a test Correlation: Causation: YES/NO