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Chapter 5
5.1 Remember: The graph of an equation in slope intercept form is: y = mx +b
Write an equation of the line with the given slope and y-intercept. 1. Slope is 8; y-intercept is 7 2. Slope is 3/4; y-intercept is -3
If you know the point where a line crosses the y-axis and any other point on the line, you can write an equation of the line.
Write the equation of the line shown. 1. Calculate the slope. 2. Notice the y-intercept
5.2 Remember: – The graphs of linear functions are lines. You can use slope-intercept form to write a linear function.
Write an equation for the linear function f with the values of f(0) = 5 and f(4) = 17 • Write the ordered pairs. • Calculate the slope • Write the equation of the line
Writing an equation of a line in SlopeIntercept Form 1. Identify the slope m. You can use the slope formula to calculate the slope if you know two points on the line. 2. Find the y-intercept. You can substitute the slope and the coordinates of a point (x, y) on the line in y = mx + b. Then solve for b. 3. Write an equation using y = mx +b.
Examples: 1. Write an equation that passes through (6, 3) and has a slope of 2. 2. Write an equation of the line that passes through (-2, 5) and (2, -1).
5.3 The point-slope form of the equation of the nonvertical line through a given point (x1, y1) with a slope of m is y - y1 = m(x - x1)
Examples: 1. Write an equation in point-slope formula of the line that passes through the point (4, -3) and has a slope of 2. 2. Graph the equation y - 1 = -(x - 2)
3. Write an equation in point-slope form for the line shown.
5.4 Remember: – The linear equation Ax + By = c is in standard form, where A, B, & C are real numbers and A and B are not both zero. All linear equations can be written in standard form.
Write two equations in standard form that are equivalent to 2x - 6y = 4 Write an equation in standard form of the line shown. 1. Calculate the slope. 2. Write an equation in point-slope form. 3. Rewrite the equation in standard form.
Remember: – Equations of horizontal lines have the form y = a. – Equations of vertical lines have the form x = b. – You CANNOT write an equation for a vertical line in slope-intercept form or slope form because a vertical line has no slope. – You CAN write an equation for a vertical line in standard form.
Examples: – Write equations of the horizontal and vertical lines that pass through the given point (-8, -9).
Examples: – Find the missing coefficient in the equation of the line shown. Write the completed equation (in standard form). • Find the value of A by substituting the coordinates of the given point for x and y.
– Find the missing coefficient in the equaton of the line that passes through (2, 11). Ax + y = -3
5.5 If two nonvertical lines in the same plane have the same slope then they are parallel. If two nonvertical lines in the same plane are parallel, then they have the same slope.
Example – Write an equation of the line that passes through (-3, -5) and is parallel to the line y = 3x - 1 • Step 1: Identify the slope. • Step 2: Find the y-intercept using the given point. • Step 3: Write an equation. Use y = mx + b
Two lines in the same plane are perpendicular if they intersect to form a right angle. – Horizontal and vertical lines are perpendicular to each other.
Perpendicular Lines If two nonvertical lines in the same plane have slopes that are negative reciprocals, then the lines are perpendicular. If two nonvertical lines in the same plane are perpendicular, then their slopes are negative reciprocals.
Example Determine which lines, if any, are parallel or perpendicular Line a: Line b: Line c: y = 5x - 3 x + 5y = 2 -10y - 2x = 0
Example Write an equation of the line that passes through (4, -5) and is perpendicular to the line y = 2x +3. – Identify the slope. Find the negative reciprocal. – Find the y-intercept. Use the slope and the given point. – Write an equation.
5.6 A scatter plot is a graph used to determine whether there is a relationship between paired data. Scatter plots can show trends in data.
When data show a positive or negative correlation, you can model the trend in the data using line of fit. 1. Make a scatter plot. 2. Decide whether the data can be modeled by a line 3. Draw a line that appears to fit the data closely. (approx. as many points above or below the line) 4. Write an equation using two points on the line. (The points don’t have to be actual data pairs, but must lie on the line of fit.)
Example The table shows the number of active redcockaded woodpecker clusters in a part of the De Soto National Forest in Mississippi. Write an equation that models the number of active clusters as a function of the number of years since 1990.
5.7 The line that most closely follows a trend in data is called the best-fitting line. The process of finding the best-fitting line to model a set of data is called linear regression. Using a line or its equation to approximate a value between two known values is called linear interpolation.
Using a line or its equation to approximate a value outside the range of known values is called linear extrapolation. Example: – Use the equation from Example 1 to approximate the number of C singles shipped in 1998 and in 2000.
A zero of a function y = f(x) is an x-value for which f(x) = 0 (or y= 0) – Because y = 0 along the x-axis of the coordinate plane, a zero of a function is an x-intercept of the function’s graph
5.1 Remember: The graph of an equation in slope intercept form is: y = mx +b
Write an equation of the line with the given slope and y-intercept. 1. Slope is 8; y-intercept is 7 2. Slope is 3/4; y-intercept is -3
If you know the point where a line crosses the y-axis and any other point on the line, you can write an equation of the line.
Write the equation of the line shown. 1. Calculate the slope. 2. Notice the y-intercept
5.2 Remember: – The graphs of linear functions are lines. You can use slope-intercept form to write a linear function.
Write an equation for the linear function f with the values of f(0) = 5 and f(4) = 17 • Write the ordered pairs. • Calculate the slope • Write the equation of the line
Writing an equation of a line in SlopeIntercept Form 1. Identify the slope m. You can use the slope formula to calculate the slope if you know two points on the line. 2. Find the y-intercept. You can substitute the slope and the coordinates of a point (x, y) on the line in y = mx + b. Then solve for b. 3. Write an equation using y = mx +b.
Examples: 1. Write an equation that passes through (6, 3) and has a slope of 2. 2. Write an equation of the line that passes through (-2, 5) and (2, -1).
5.3 The point-slope form of the equation of the nonvertical line through a given point (x1, y1) with a slope of m is y - y1 = m(x - x1)
Examples: 1. Write an equation in point-slope formula of the line that passes through the point (4, -3) and has a slope of 2. 2. Graph the equation y - 1 = -(x - 2)
3. Write an equation in point-slope form for the line shown.
5.4 Remember: – The linear equation Ax + By = c is in standard form, where A, B, & C are real numbers and A and B are not both zero. All linear equations can be written in standard form.
Write two equations in standard form that are equivalent to 2x - 6y = 4 Write an equation in standard form of the line shown. 1. Calculate the slope. 2. Write an equation in point-slope form. 3. Rewrite the equation in standard form.
Remember: – Equations of horizontal lines have the form y = a. – Equations of vertical lines have the form x = b. – You CANNOT write an equation for a vertical line in slope-intercept form or slope form because a vertical line has no slope. – You CAN write an equation for a vertical line in standard form.
Examples: – Write equations of the horizontal and vertical lines that pass through the given point (-8, -9).
Examples: – Find the missing coefficient in the equation of the line shown. Write the completed equation (in standard form). • Find the value of A by substituting the coordinates of the given point for x and y.
– Find the missing coefficient in the equaton of the line that passes through (2, 11). Ax + y = -3
5.5 If two nonvertical lines in the same plane have the same slope then they are parallel. If two nonvertical lines in the same plane are parallel, then they have the same slope.
Example – Write an equation of the line that passes through (-3, -5) and is parallel to the line y = 3x - 1 • Step 1: Identify the slope. • Step 2: Find the y-intercept using the given point. • Step 3: Write an equation. Use y = mx + b
Two lines in the same plane are perpendicular if they intersect to form a right angle. – Horizontal and vertical lines are perpendicular to each other.
Perpendicular Lines If two nonvertical lines in the same plane have slopes that are negative reciprocals, then the lines are perpendicular. If two nonvertical lines in the same plane are perpendicular, then their slopes are negative reciprocals.
Example Determine which lines, if any, are parallel or perpendicular Line a: Line b: Line c: y = 5x - 3 x + 5y = 2 -10y - 2x = 0
Example Write an equation of the line that passes through (4, -5) and is perpendicular to the line y = 2x +3. – Identify the slope. Find the negative reciprocal. – Find the y-intercept. Use the slope and the given point. – Write an equation.
5.6 A scatter plot is a graph used to determine whether there is a relationship between paired data. Scatter plots can show trends in data.
When data show a positive or negative correlation, you can model the trend in the data using line of fit. 1. Make a scatter plot. 2. Decide whether the data can be modeled by a line 3. Draw a line that appears to fit the data closely. (approx. as many points above or below the line) 4. Write an equation using two points on the line. (The points don’t have to be actual data pairs, but must lie on the line of fit.)
Example The table shows the number of active redcockaded woodpecker clusters in a part of the De Soto National Forest in Mississippi. Write an equation that models the number of active clusters as a function of the number of years since 1990.
5.7 The line that most closely follows a trend in data is called the best-fitting line. The process of finding the best-fitting line to model a set of data is called linear regression. Using a line or its equation to approximate a value between two known values is called linear interpolation.
Using a line or its equation to approximate a value outside the range of known values is called linear extrapolation. Example: – Use the equation from Example 1 to approximate the number of C singles shipped in 1998 and in 2000.
A zero of a function y = f(x) is an x-value for which f(x) = 0 (or y= 0) – Because y = 0 along the x-axis of the coordinate plane, a zero of a function is an x-intercept of the function’s graph
