Sect 3.1 Reading Graphs How much money will you earn in a lifetime with an associate’s degree? What degree must you obtain to earn at least 2 million dollars in a lifetime? $1.5 to 1.6 million Bachelor’s or higher.

Sect 3.1 Reading Graphs Find the amount of billion that was given to Pell Grants.

Sect 3.1 Reading Graphs How many months of regular exercise are required to lower the pulse rate as much as possible? How many months of regular exercise are needed to achieve a pulse rate of 65 beats per min.? Indicates no visual comparisons! 6 MONTHS 3 MONTHS

Sect 3.1 Reading Graphs All points are labeled (x, y). x y

Sect 3.1 Reading Graphs These points are not in a Quadrant!

Sect 3.2 Graphing Linear Equations Determine if (2, 5) and (-1, -1) are solutions to TRUE Yes (2, 5) is a solution. TRUE Yes (-1, -1) is a solution. (2, 5) (-1, -1)

Sect 3.2 Graphing Linear Equations Graph x (x, y) When the equation is solved for y, y is the dependent variable and x is the independent variable. This means we can pick values for x and substitute them in to find the y value. Start with x = 0. Easiest to multiply by. Use a straight edge to connect the points to form a straight line.

Sect 3.2 Graphing Linear Equations Graph x (x, y) Start with x = 0. Easiest to multiply by. Use a straight edge to connect the points to form a straight line.

Sect 3.2 Graphing Linear Equations Graph x(x, y) Start with x = 0, Count by 3’s due to the denominator in the fraction. Notice a pattern in the points? Y-coordinates and X-coordinates The fraction contains the changes in the y-coord. and x-coord.!

Sect 3.2 Graphing Linear Equations Graph x(x, y) Start with x = 0, Count by 2’s due to the denominator in the fraction. Use a straight edge to connect the points to form a straight line. +2

Sect 3.2 Graphing Linear Equations Graph x(x, y) Solve the equation for y! + 5y + 5y

Sect 3.3 Graphing Linear Equations in Standard Form Graph x-intercept ( ____, 0 ) y-intercept ( 0, ____ )

Sect 3.3 Graphing Linear Equations in Standard Form Graph x-intercept ( ____, 0 ) y-intercept ( 0, ____ )

Sect 3.3 Graphing Linear Equations in Standard Form Graph x-intercept ( ____, 0 ) y-intercept ( 0, ____ )

Sect 3.3 Graphing Linear Equations in Standard Form Graph x-intercept ( ____, 0 ) y-intercept ( 0, ____ )

Sect 3.3 Graphing Linear Equations in Standard Form Graph x-intercept ( ____, 0 ) y-intercept ( 0, ____ ) Rewrite in standard form: Ax + By = C Short Cut: Cover-up technique. To find the x-intercept, cover-up the y term. Solve for x. To find the y-intercept, cover-up the x term. Solve for y.

Sect 3.3 Graphing Linear Equations Graph x-intercept ( ____, 0 ) y-intercept ( 0, ____ ) Notice there is no y variable in the equation…the line can’t cross the y-axis. No y-intercept.

Sect 3.3 Graphing Linear Equations Graph x-intercept ( ____, 0 ) y-intercept ( 0, ____ ) Notice there is no x variable in the equation…the line can’t cross the x-axis. No x-intercept.

Sect 3.3 Graphing Linear Equations Graph x-intercept ( ____, 0 ) y-intercept ( 0, ____ ) YUCK! Lets find clean points. Take the LARGEST coefficient and take it’s opposite to the other side. Start with 11 and keep subtracting by 5 until you have a number divisible by 3. We only subtracted by one 5, so x = 1.

Sect 3.3 Graphing Linear Equations Graph x-intercept ( ____, 0 ) y-intercept ( 0, ____ ) YUCK! Lets find clean points. Take the LARGEST coefficient and take it’s opposite to the other side. Start with 19 and keep subtracting by 5 until you have a number divisible by 5. We subtracted by two 7’s, so x = 2.

Sect 3.4 Rates A Rate is a ratio that indicates how two quantities change with respect to each other. Unit rate is when the second quantity is one. On Jan. 3 rd, Joe rented a car with a full tank of gas and 9312 miles on the odometer. On Jan. 7 th, he returned the car with 9630 miles on the odometer. If Joe had to pay $108 for the total bill which included 12 gallons of gas, find the following rates. Convert to unit rates. 1. Gas consumption in miles per gallon. 2. Average cost of the rental in dollars per day. 3. Travel rate in miles per day. Divide for unit rate. Jan. 3 rd, 4 th, 5 th, 6 th, and 7 th = 5 days

Sect 3.5 Slope The Slope is a ratio that indicates how the change in the y-coordinates change with the respect to the change in the x-coordinates. The slope of a line contains two points ( x 1, y 1 ) and ( x 2, y 2 ) is given by Change in y Change in x

Sect 3.5 Graphing Linear Equations Graph ( -4, 3 ) and ( 2, -6 ) and find the slope. Change in y = -9 Change in x = 6

Sect 3.5 Graphing Linear Equations Positive slopes always have lines that go uphill. Slopes > 1 are steep. 0 < Slopes < 1 begin to flatten out.

Sect 3.5 Graphing Linear Equations Negative slopes always have lines that go downhill. Slopes < -1 are steep. -1 < Slopes < 0 begin to flatten out.

Sect 3.5 Graphing Linear Equations Graph ( -4, 3 ) and ( 2, 3 ) and find the slope. Horizontal Line Graph ( 5, -1 ) and ( 5, 6 ) and find the slope. Vertical Line

Sect 3.5 Slope The Grade is a slope that is measured as a percent. Drop 7 feet for every 100 feet traveled horizontally.

Sect 3.6 Graphing Linear Equations in Slope Intercept Form Graph ( 0, - 4 ) Starting point Directions 2 up & 3 right Or opposite 2 down & 3 left

Sect 3.6 Graphing Linear Equations in Slope Intercept Form Graph ( 0, 6 ) Starting point Directions 5 down & 2 right + y = + y ( 0, -3 ) Starting point Directions 2 up & 1 right Opposite 2 down & 1 left -3

Sect 3.6 Graphing Linear Equations in Slope Intercept Form Graph Solving for y would not be a good decision because it will generate a fraction for a y-intercept. x-intercept and slope. x-intercept and y-intercept are bad…fractions. x = 3 for the three 5’s we subtracted.

Sect 3.6 Graphing lines Parallel Lines – Lines are parallel when they have the same slopes or the lines are vertical. Perpendicular Lines – Lines are perpendicular when their slopes are opposite reciprocals of each other. I prefer to say, “the perpendicular slopes are opposite reciprocals.”

Sect 3.7 Graphing Linear Equations in Point Slope Form Consider a line with slope 2 passing through the point ( 4, 1 ). ( 4, 1 ) ( x, y ) Consider removing the “m” and re-writing so there is no fraction. The equation is called point-slope form of a linear equation with the slope m and the point. Notice the sign change!

Sect 3.7 Graphing Linear Equations in Point Slope Form Graph (1, 5) m = -3 4 (-4, 2) m = 1313 (-4, -5) m = 3232 (2, -5) m = 4141

Sect 3.7 Graphing lines Write the equation of a line in point-slope form with the slope of and the point ( -14, 12). Convert to slope-intercept form. Write the equation of a line in point-slope form with the slope of and the point ( 4, -6). Convert to slope-intercept form = + 12 – 6 = – 6

Sect 3.7 Graphing lines…ANOTHER APPROACH Write the equation of a line in point-slope form with the slope of and the point ( 4, -6). Convert to slope-intercept form. NO FRACTION WORK…Remember Standard Form. Substitute the point ( 4, -6 ) for x and y to solve for C. – 8x = – 8x Solve for y. Short cut.

Sect 3.7 Graphing lines Write the equation of a line that is parallel to 3x + 5y = 11 and contains the point ( 0, 2). Use slope-intercept form. Write the equation of a line that is perpendicular to 3x + 5y = 11 and contains the point ( 0, 2). Use slope-intercept form. y-intercept (0, b) Same slopes. Opposite and Reciprocal slopes. Perpendicular slope. y-intercept (0, b)

Sect 3.7 Graphing lines Write the equation of a line that is parallel to 7x – 4y = 9 and contains the point ( 5, -3). Use slope-intercept form. Write the equation of a line that is perpendicular to 8x + 3y = 13 and contains the point ( -7, 4). Use slope-intercept form. Same slopes. Must be the same!. Perpendicular slope. The numbers must flip and change the sign! New.

Sect 9.4 Graphing Linear Inequalities Graph the line. Consider changing the equation to an inequality. Determine if the points are solutions, ( -4, 3 ), (-8, 5 ), and ( 2, -1 ). FALSETRUE

Sect 9.4 Graphing Linear Inequalities in Slope Intercept Form Graph The inequality has y > which tells me to shade all the y coordinates greater than the line…above the line. – 2x = – 2x Solve for y. Remember to flip the inequality symbol. Shade all the y coordinates less than the line…below the line. No equal to line means a dashed line! Equal to line means a solid line!

Sect 9.4 Graphing Linear Equations Graph the line. Consider changing the equation to an inequality. Determine which way to shade. Test the Origin (0, 0) TRUE Shade in the direction of the Origin Find the x and y intercepts.

Sect 9.4 Graphing Linear Equations in Standard Form Graph Find the x and y intercepts. Test the Origin (0, 0) TRUE Find the x and y intercepts. Test the Origin (0, 0) FALSE

Sect 9.4 Graphing Linear Equations Graph

Sect 9.4 Graphing Linear Equations Graph