Exercise 4 Find the value of k such that the line passing through the points (−4, 2k) and (k, −5) has slope −1.

Objective 2 You will be able use slope to be able to tell what kind of line you have

The Slope Game The slope of a line indicates whether it rises or falls (L to R) or is horizontal or vertical. m > 0 m < 0 m = 0 m = undef Insert Picture As the absolute value of the slope of a line increases, --?--. the line gets steeper.

Exercise 5 Without graphing, tell whether the line through the given points rises, falls, is horizontal, or is vertical. (1, 6); (8, −1) (−4, −3); (7, 1) (−5, 3); (−5, 1) (9, 2); (−9, 2)

You will be able to find the slopes of parallel and perpendicular lines Objective 3

Parallel and Perpendicular Two lines are parallel lines iff they are coplanar and never intersect. Two lines are perpendicular lines iff they intersect to form a right angle. 𝑚∥𝑛

Parallel and Perpendicular Two lines are parallel lines iff they have the same slope. Two lines are perpendicular lines iff their slopes are negative reciprocals.

Exercise 6 Tell whether the pair of lines are parallel, perpendicular, or neither Line 1: through (-2, 1) and (0, -5) Line 2: through (0, 1) and (-3, 10) Line 1: through (-2, 2) and (0, -1) Line 2: through (-4, -1) and (2, 3)

Exercise 7 If two distinct lines are parallel, what do you know about their y-intercepts? If one of two perpendicular lines has a slope of 1/a and a < 0, is the slope of the other line positive or negative?

Objective 4 You will be able to graph the equation of a line in slope- intercept or standard form

Family of Linear Functions Linear Parent Function Parent Functions Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function. Family of Linear Functions Linear Parent Function 𝑦=−𝑥 𝑦=𝑥 𝑦=3𝑥–7 𝑦= 1 2 𝑥−9

Family of Quadratic Functions Quadratic Parent Function Parent Functions Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function. Family of Quadratic Functions Quadratic Parent Function 𝑦=− 𝑥 2 𝑦= 𝑥 2 𝑦=3 𝑥 2 –7 𝑦= 1 2 𝑥 2 −9

Parent Functions Who is the simplest member of your family? Well, in math, the simplest member of a family of functions is called the parent function. Family of Functions Parent Function A group of functions that share common characteristics Simplest member of the family

Linear parent function: 𝑦=𝑥 or 𝑓(𝑥)=𝑥 Parent Functions Linear parent function: 𝑦=𝑥 or 𝑓(𝑥)=𝑥 All other linear functions can be formed with transformations on the parent function.

Intercepts The 𝒙-intercept of a graph is where it intersects the 𝑥-axis. 𝑎,0 0,𝑏 The 𝒚-intercept of a graph is where it intersects the 𝑦-axis. Click me!

Slope-Intercept Slope-Intercept Form of a Line: If the graph of a line has slope 𝑚 and a 𝑦-intercept of (0, 𝑏), then the equation of the line can be written in the form 𝑦=𝑚𝑥+𝑏 Slope 𝑦-intercept

Use 𝑚 to plot more points Slope-Intercept To graph an equation in slope-intercept form: Plot 0,𝑏 2. Solve for 𝑦 Use 𝑚 to plot more points 1. 3. Draw line 4.

Exercise 8a Without your graphing calculator, graph each of the following: 𝑦=−𝑥+2 𝑦= 2 5 𝑥+4

Exercise 8b 𝑓(𝑥)=1–3𝑥 8𝑦=−2𝑥+20 Without your graphing calculator, graph each of the following: 𝑓(𝑥)=1–3𝑥 8𝑦=−2𝑥+20

Standard Form Standard Form of a Line The standard form of a linear equation is 𝐴𝑥+𝐵𝑦=𝐶, where 𝐴 and 𝐵 are not both zero. Generally taken to be integers

Standard Form To graph an equation in standard form: Write equation in standard form. Let x = 0 and solve for y. This is your y-intercept. Let y = 0 and solve for x. This is your x-intercept. Connect the dots.

Standard Form Let 𝑥=0 Let 𝑦=0 Draw line To graph an equation in standard form: Let 𝑥=0 Let 𝑦=0 Draw line 1. 2. 3. Solve for 𝑦 Solve for 𝑥 This is the 𝑦-intercept This is the 𝑥-intercept

Exercise 9a Without your graphing calculator, graph each of the following: 2𝑥+5𝑦=10 −2𝑥−3𝑦=6

Exercise 9b Without your graphing calculator, graph each of the following: 3𝑥−2𝑦=12 8𝑦=−2𝑥+20

Exercise 10 For an equation in standard form, Ax + By = C, what is the slope of the line in terms of A and B?

Horizontal and Vertical Lines Horizontal Line The graph of 𝑦=𝑏 is a horizontal line through (0,𝑏). Vertical Line The graph of 𝑥=𝑎 is a vertical line through (𝑎,0).

Exercise 11 Graph each of the following: 𝑥=1 𝑦=−4

Find Slope & Rate of Change Graph Equations of Lines 2.2: Slope; 2.3: Graph Lines Find Slope & Rate of Change Graph Equations of Lines Objectives: To find the slope of a line given 2 points To classify a line based on its slope To find the slope of parallel and perpendicular lines To graph the equation of a line using slope-intercept and standard form of a line

Objective 1 You will be able to use a graphing calculator to graph a scatter plot and to find a line of best fit

If 𝑦 increases as 𝑥 increases, there is a positive correlation. Let’s say a set of data consists of two quantities, 𝑥 and 𝑦. In statistics, a correlation exists between 𝑥 and 𝑦 if there is a linear relation between 𝑥 and 𝑦. If 𝑦 increases as 𝑥 increases, there is a positive correlation.

If 𝑦 decreases as 𝑥 increases, there is a negative correlation. Let’s say a set of data consists of two quantities, 𝑥 and 𝑦. In statistics, a correlation exists between 𝑥 and 𝑦 if there is a linear relation between 𝑥 and 𝑦. If 𝑦 decreases as 𝑥 increases, there is a negative correlation.

If there’s no obvious pattern, there is approximately no correlation. Let’s say a set of data consists of two quantities, 𝑥 and 𝑦. In statistics, a correlation exists between 𝑥 and 𝑦 if there is a linear relation between 𝑥 and 𝑦. If there’s no obvious pattern, there is approximately no correlation.

Exercise 5 Describe the correlation shown by each scatter plot.

Correlation Coefficient A correlation coefficient for a set of data measures the strength of the correlation. -1 ≤ r ≤ 1 Perfect negative correlation = -1 Perfect positive correlation = 1 No correlation = 0

Correlation Coefficient A correlation coefficient for a set of data measures the strength of the correlation. -1 ≤ r ≤ 1

Exercise 6 The table shows the number 𝑦 (in thousands) of alternative-fueled vehicles in use in the United States 𝑥 years after 1997. Graph this data as a scatter plot. Determine if a correlation exists.

Graphing Calculator Instructions Entering the data: Press the STAT key and choose 1:Edit… Under the list L1, enter all of the 𝑥-values, hitting enter between values. Press the right arrow key, and under the list L2, enter all of the 𝑦-values, hitting enter between values. Take few seconds to check for typos.

Graphing Calculator Instructions Graphing the data: Press 2ND then the Y= key to access the STAT PLOT menu. Choose your favorite Plot#, press ENTER. Turn Plot On. Choose the scatter plot icon. Make sure the 𝑥’s come from L1 and the 𝑦’s come from L2. Choose your favorite Mark.

Graphing Calculator Instructions Graphing the data: Press the ZOOM key. Choose 9:ZoomStat. This chooses a good viewing rectangle (domain and range) based on the values entered in L1 and L2 Enjoy

Line of Best-Fit If a strong correlation exists between x and y, where | r | is near 1, then the data can be reasonable modeled by a trend line.

Line of Best-Fit This line of best fit lies as close as possible to all the data points, with as many above as below.

Exercise 7 Find the best-fitting line from Exercise 6.

Graphing Calculator Instructions Finding the trend line: Press 2ND then the 0 (zero) key to access the CATALOG menu. Press the “D” key (𝑥-1). Scroll down and press ENTER on DiagnosticOn. Press ENTER again. You only have to do this once. Doing this will return the r- and r2-values for a trend line.

Graphing Calculator Instructions Finding the trend line: Press the STAT key. Use the right arrow key to move over to the CALC menu. Choose 4:LinReg(ax+b). This is called a linear regression, and it will return the trend line in slope-intercept form. Now you’re back on the HOME screen. Now we have to tell it where to find the data.

Graphing Calculator Instructions Finding the trend line: Press 2ND “1” for the L1 key. These are your 𝑥-values. Press the , (comma) key. Press 2ND “2” for the L2 key. These are your 𝑦-values.

Graphing Calculator Instructions Finding the trend line: Press VARS key to access the variables menu. Use the right arrow key to scroll over to the Y-VARS menu. Choose 1:Function… Choose 1:Y1. Press ENTER.

Graphing Calculator Instructions Finding the trend line: The HOME screen now shows you the values for a (slope), b (y-intercept), r2 (r2), and r (the correlation coefficient). To view the trend line, press the GRAPH key.

Graphing Calculator Instructions To evaluate a trend line at a data point: On the HOME screen, input Y1 (VARS > Y-VARS > Functions… > Y1) In parentheses, enter your chosen 𝑥-value: Y1(100) for example. Press ENTER.

Exercise 8 Use your line of best fit to predict the number of alternative-fueled vehicles in use in the United States 14 years after 1997.

Exercise 9 The table gives the systolic blood pressure 𝑦 of patients 𝑥 years old. Determine if a correlation exists. If it is a strong correlation, find the line of best fit. Predict the systolic blood pressure of a 16-year-old.

Exercise 9 According to your model Y(16)=96, but: Girls (Age 16) Boys (Age 16) 122-132 125-138 Our prediction was so far off because you can’t reliably extrapolate this far away from the data

2.6: Draw Scatter Plots & Best-Fitting Lines 2.5: Direct Variation; 2.6: Scatter Plots 2.6: Draw Scatter Plots & Best-Fitting Lines Objectives: To model direct variation To use a graphing calculator to draw a scatter plot and a best-fitting line