Name: Teacher: Per: Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10 Unit 3a [Graphing Linear Equations] Unit 2 Solving Equations 1
To be a Successful Algebra class, TIGERs will show #TENACITY during our practice, have I attempt all practice I attempt all homework I never give up when I don t understand #INTEGRITY as we help others with their work, maintain a #GO-FOR-IT attitude, continually I always check my answers I correct my work, I never just copy answers I explain answers, I never just give them I write down all notes, even if I m confused I remain positive about my goals I treat each day as a chance to reset #ENCOURAGE each other to succeed as a team, and always #REACH-OUT and ask for help when we need it! I offer help when I understand the material I push my teammates to reach their goals I never let my teammates give up I ask my questions during homework check I ask my teammates for help during practice I attend enrichment/tutorials when I need to Unit 2 Solving Equations 2
Unit Calendar MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY October 6 7 8 9 10 Graphing from a Table Slope Slope QUIZ Slope Intercept Form Parent Function And Transformations 13 14 15 16 17 Student Holiday Slope Intercept Form QUIZ PSAT Standard Form and Intercepts Standard to Slope Intercept Form 20 21 22 23 24 Solving for Slope Intercept Form Review TEST Essential Questions What does it mean for relationship to be a linear function? How can linear functions be used to model problem situations? What are the critical features of a graph and how can they help me solve this problem? How can a value of a function be represented? What is the meaning of slope and how do I find it in this situation? Unit 2 Solving Equations 3
Critical Vocabulary Linear Function Parent Function Slope x-intercept / Zero y-intercept Direct Variation Parallel Perpendicular Unit 2 Solving Equations 4
Graphing from a Table Examples: Equation: y = 3x 5 Table x y -2-1 0 1 2 3 Graph Equation: y = 2x + 6 Table Graph x y -2-1 0 1 2 3 To babysit my younger brother, my mom gives me $5 and an extra $2 per hour Equation: y = Table x y -2-1 0 1 2 3 Graph Equation: y = 2 3 x + 5 Table x y -6-3 0 3 6 9 Graph Unit 2 Solving Equations 5
Practice: Equation: y = 2x 4 x -2-1 0 1 2 3 y With no money in my bank, I started saving $3 every week Equation: y = x -2-1 0 1 2 3 y Equation: y = 4x + 3 x -2-1 0 1 2 3 y My box of chocolates had 5 pieces in it I ate one piece each class period Equation: y = x -2-1 0 1 2 3 y Equation: y = 1 3 x 4 x -6-3 0 3 6 9 y Equation: y = 3 2 x + 4 x -4-2 0 2 4 6 y Unit 2 Solving Equations 6
Unit 2 Solving Equations 7
Slope The slope of a line is the measure of how the line is Slope is directly related to the in a rule Slope Formula: m = y 2 y 1 x 2 x 1 Change in y Change in x Rise Run Positive Negative Zero (0) Undefined Increases from Left to Right Decreases from Left to Right Stays the same (constant) from Left to Right Not a function, so no definition for slope Unit 2 Solving Equations 8
Examples: Slope Formula: m = y 2 y 1 x 2 x 1 Table 2 - Points Graph Situation x y -2 12-1 10 0 8 1 6 2 4 ( -2, 12 ) and ( 1, 6 ) I have $8 to spend on snacks during the week Each snack I purchase costs $2 x y -4-1 -2 2 0 5 2 8 4 11 ( 0, 5 ) and (-2, 2 ) Every 2 months I have to pay a $3 charge to keep my account open Originally, I had to pay a $5 fee to open the account x y -2 4-1 4 0 4 1 4 2 4 ( -1, 4 ) and (2, 4 ) A solar powered go cart began moving at 4 mph and never changed speed x y -3-2 -3-1 -3 0-3 1-3 2 ( -3, -2 ) and ( -3, 1 ) N/A This is not a function, so it does not have a situation to match it Unit 2 Solving Equations 9
Practice: x y -4-14 -2-7 0 0 2 7 4 14 ( 1, 3 ) and ( 7, 6 ) I am paying my friend back $4 every week from the $20 I borrowed from her m = m = m = m = x Y 1-2 2-2 3-2 4-2 5-2 ( 0, 0 ) and ( 9, -6 ) x y -2-3 -1 2 0 7 1 12 2 17 m = m = m = m = Benjamin got a $40 gift card to Starbucks for his birthday His favorite drink costs $3 each time he goes ( 4, -1 ) and ( 4, 9 ) Every 6 months I pay my car insurance $131, plus I paid an intial fee of $25 when I signed up m = m = m = m = x y -3-4 -2-3 -1-2 0-1 1 0 ( 4, -5 ) and ( -2, -7 ) x y -5 17-3 11-1 5 1-1 3-7 m = m = m = m = Unit 2 Solving Equations 10
Unit 2 Solving Equations 11
Slope Intercept Form Slope-Intercept Form: The equation of a line written in a form that includes both the and the - (m) (b) Examples: y = 3x 5 y = x + 7 y = 4x y = 1 4 x 5 y = 3 2 x + 7 y = 2 5 x Unit 2 Solving Equations 12
Special Cases: Horizontal and Vertical Lines H O Y Horizontal Line Zero Slope 0 5 y = constant y = 2 V Vertical Line U Undefined Slope 3 0 X x = constant x = 4 Examples: y = 2 x = 2 y = 2x Unit 2 Solving Equations 13
Practice: y = 2x + 3 y = 2 3 x + 1 y = 6 y = 3x y = x 5 y = 4 3 x 1 x = 7 y = 3 2 x y = 4x + 9 Unit 2 Solving Equations 14
Linear Parent Function and Transformations Transformations: The parts of an equation that change the shape or position of a graph/line Slope / Steepness: Ignoring the negative, if the slope is greater than one then the line is steeper y = x y = 4x y = 5 2 x **Improper Fractions Ignoring the negative, if the slope is less than one then the line is less steep y = x y = 1 2 x **Proper Fractions Unit 2 Solving Equations 15
Tran SL ation: SL ide If there is a constant (regular number with no variable), the change is a vertical shift or slide of the line y = 1 2 x y = 1 2 x + 3 Re FL ection: FL ip If the coefficient (number in front) of x is negative, the change is a vertical flip of the line y = 1 2 x + 3 y = 1 2 x + 3 Parent Function: The most basic form of a graph (No Transformations) Linear Parent Function: y = x y = +1x + 0 No Reflection, Slope of 1, No Translation Unit 2 Solving Equations 16
Examples: y = 4x 3 Steepness: More Steep Less Steep No Change Translation: Shift Up Shift Down No Change Reflection: Reflection No Change Parent Function Transformed Equation Steepness: More Steep Less Steep No Change y = 2 5 x + 1 Translation: Shift Up Shift Down No Change Reflection: Reflection No Change Parent Function Transformed Equation Equation Steepness Translation Reflection y = 4 3 x + 5 y = 3x y = x 27 Unit 2 Solving Equations 17
Practice: y = 4 3 x + 1 Steepness: More Steep Less Steep No Change Translation: Shift Up Shift Down No Change Refelction: Reflection No Change Parent Function Transformed Equation Equation Steepness Translation Reflection y = 6x + 5 y = 2 7 x 1 y = x y = 3x 8 y = 3 2 x 6 y = x + 4 y = 1 4 x y = x 35 Unit 2 Solving Equations 18
Unit 2 Solving Equations 19
Slope Intercept Form (Applications) Examples: I owe my friend $10 and I plan to pay them back $2 each day The equation that models this situation is y = 2x 10 A) What does the y represent? B) What does the x represent? C) What does the 2 represent? D) What does the 10 represent? E) Why is the 10 negative in this problem? G) Graph the equation F) Would this be continuous or discrete? H) What is a reasonable domain for this problem? Gina has a $30 gift card to Starbuck s Every time she buys a coffee she is charged $3 A) Write an equation in Slope-Intercept Form to represent the problem B) Why did you choose the sign of 3 to be positive or negative? C) If we doubled the constant in the equation, what would that represent in the problem? D) What is a reasonable domain for this situation? E) Is this situation discrete or continuous? Unit 2 Solving Equations 20
Practice: Chris does a lot of babysitting When parents drop off their children and Chris can supervise at home, the hourly rate is $3 If Chris has to travel to the child s home, there is a fixed charge of $5 for transportation in addition to the $3 hourly rate A) Write an equation for when he watches the kids at home? B) Write an equation for when he needs to travel to watch the kids? C) Graph both Lines D) Will a parent dropping off their kid ever pay more than if Chris has to travel to their home? E) If in 3 years, Chris has a new equation, y = 5x + 10, what has changed in this problem? The charges for Anderson s Plumbing can be modeled by the following equation, where, C, is the total cost for plumbing services that last, h, hours: C 40 25h A) What does the 25 represent in this situation? B) What does the 40 represent in this situation? The cost of dinner at a sweet 16 party is $300 plus $10 per person A) What is the equation that models this situation? B) What does the y-intercept represent? C) What does the slope represent? D) Find the cost for 50 guests Unit 2 Solving Equations 21
Laura is on a 2 day hike in the Smoky Mountains She hiked 8 miles the first day and is hiking at a rate of 3 mi/h on the second day Her total distance is a function of the time she hikes A) What is the equation that models this situation? B) What does the y-intercept represent? C) What does the slope represent? D) What will be Laura s total distance if she hikes 6 hours on the second day? The equation for a car driving away from Katy is represented by the equation: y 46x, where y is the distance from Katy after x hours A) What does y represent in this problem? B) What does x represent in this problem? C) What is the meaning of the slope? D) What would the slope and y-intercept in the equation y 46x 184 represent? The Star Car Rental Company charges a flat fee of $30 plus $025 per mile to rent a car A) Which value would be considered the y-intercept or beginning value? B) Which value would be considered the slope or rate of change? C) Write the equation for this situation D) If You only had $50 to spend on the rental car, how many miles could you drive? Unit 2 Solving Equations 22
Unit 2 Solving Equations 23
Standard Form and Intercepts Standard Form: The equation of a line written in the form Ax + By = C A is always positive and C is a constant x - intercept: The point where a graph touches the x axis (, 0 ) y - intercept: The point where a graph touches the y axis ( 0, ) Examples: Equation: 3x + 5y = 30 x intercept: (, 0 ) y intercept: ( 0, ) Equation: 4x 6y = -12 Unit 2 Solving Equations 24
Practice: Equation: 2x + 8y = -16 Equation: 5x 10y = 20 Equation: 9x + 3y = 18 Equation: 10x 5y = -30 Equation: 4x + 6y = -24 Equation: 8x 8y = 32 Unit 2 Solving Equations 25
Solving Standard Form for Slope Intercept Form Examples: Equation: 9x + 3y = 18 Solve for Slope-Intercept Form: 9x + 3y = 18 Equation: 2x + 8y = -8 Solve for Slope-Intercept Form: 2x + 8y = -8 Unit 2 Solving Equations 26
Practice: Equation: 5x 10y = 30 Solve for Slope-Intercept Form: 5x 10y = 30 Equation: 7x 7y = 56 Solve for Slope-Intercept Form: 7x 7y = 56 Continued Unit 2 Solving Equations 27
Equation: 4x + 6y = -12 Solve for Slope-Intercept Form: 4x + 6y = -12 Equation: 10x 5y = -40 Solve for Slope-Intercept Form: 10x 5y = -40 Unit 2 Solving Equations 28
Verifying Solutions Solution (to a line): Is ( -1, 4 ) a solution to 2y + 2 = 6x Any order pair that is a part of the line Solve and graph 2y + 2 = 6x Is ( 2, -3 ) a solution to 2x + 3y + 15 = 0 Solve and graph 2x + 3y + 15 = 0 Is ( 4, 2 ) a solution to 2y = 8 Solve and graph 2y = 8 Is ( -3, 5 ) a solution to x + 3 = 0 Solve and graph x + 3 = 0 Unit 2 Solving Equations 29
Practice: Is ( 0, -5 ) a solution to 3y + 4x = 15 Solve and graph 3y + 4x = 15 Is ( -5, -2 ) a solution to x + y = 3 Solve and graph x + y = 3 Is ( 2, 0 ) a solution to 3x + 2y 8 = 0 Solve and graph 3x + 2y 8 = 0 Is ( 2, -2 ) a solution to 6x = 12 Solve and graph 6x = 12 Continued Unit 2 Solving Equations 30
Is ( -6, -1 ) a solution to 2x + 6y = 6 Solve and graph 2x + 6y = 6 Is ( 0, -2 ) a solution to 4y 3x = 8 Solve and graph 4y 3x = 8 Is ( 2, -2 ) a solution to y 5 = 3 Solve and graph y 5 = 3 Is ( 1, 5 ) a solution to x + y = 4 Solve and graph x + y = 4 Unit 2 Solving Equations 31
Unit 2 Solving Equations 32
Unit 2 Solving Equations 33
Unit 2 Solving Equations 34