fom 11-le3-graphing linear equations review-2...

FOM 11 LE3 – GRAPHING LINEAR EQUATIONS REVIEW - 2 1

© R. Ashby 2019. Duplication by permission only.

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

1) y-INTERCEPT = the point where the graph touches or crosses the y-axis. It occurs when x = 0 .

2) x-INTERCEPT = the point where the graph touches or crosses the x-axis. It occurs when y = 0 .

CALCULATING x & y-INTERCEPTS I) An intercept is a point where the graph touches or crosses an axis, either the x or the y-axis. The y-intercept is

the point where the graph touches or crosses the y-axis. The x-intercept is the point where the graph touches or crosses the X-axis. A) THE y-INTERCEPT IS THE POINT WHERE THE GRAPH TOUCHES OR CROSSES THE Y-AXIS.

1) INVESTIGATION 1: Use the graphs found at the top of page 296 and question 4(c) of page 303 of your text to answer these questions. {Answers are on page 7 of these notes.} a) Determine the coordinates of the y-intercept of each graph.

Top of pg. 296 pg. 303 Q# 4(c)

b) What do the y-intercepts of each graph have in common?

c) What do you think is the same for every y-intercept of every graph have?

2) y-INTERCEPT USE THESE STEPS TO CALCULATE THE 1: Substitute each x variable with zero (0). 2: Solve for y.

3) SAMPLE PROBLEMS 1: Calculate the y-intercept of the graphs of these equations. Be sure you understand and memorize the process used to complete them. 1) y = 3x −2 2) y = −2x 2−x + 9 3) y = 8

3 x 3 + 5x 2−x

4) REQUIRED PRACTICE 1: Calculate the y-intercept of the graphs of these equations. SHOW YOUR PROCESS!! {Answers are on page 7 of these notes.} 1) y = x −6 2) y = 5x −10 3) y = 4

5 x −6

4) y = x 2 + 2x 5) y = −2x 2 + 4x −3 6) y = −5x 3 + 7x + 2

7) y = 3 x −6( )+ 5 8) y = 1

2x+3 9) y = x

x2+3x+2



B) THE x-INTERCEPT IS THE POINT WHERE THE GRAPH TOUCHES OR CROSSES THE X-AXIS. 1) INVESTIGATION 2: Use the graphs found at the top of page 296 and the middle of page 298 of your text to

answer these questions. {Answers are on page 7 of these notes.} a) Determine the coordinates of the x-intercept(s) of each graph.

Top of pg. 296 Middle of pg. 298

b) What do the x-intercepts of each graph have in common?

c) What do you think is the same for every x-intercept of every graph have?

2) x-INTERCEPT USE THESE STEPS TO CALCULATE THE 1: Substitute each y variable with zero (0). 2: Solve for x.

3) SAMPLE PROBLEMS 2: Calculate the x-intercept of the graphs of these equations. Be sure you understand and memorize the process used to complete them. 1) y = 3x −2 2) y = −2x + 9 3) y = 8

3 x

4) REQUIRED PRACTICE 2: Calculate the x-intercept of the graphs of these equations. SHOW YOUR PROCESS!! {Answers are on page 7 of these notes.} 1) y = x −6 2) y = 5x −10 3) y = 4

5 x −6

GRAPHING LINEAR EQUATIONS USING x & y-INTERCEPTS I) Recall that all linear equations have these three characteristics: All linear equations:

1. have two different variables that lack visible exponents. 2. have variables that are not inside a radical sign x( ) , absolute value bars y( ) , or a denominator.

3. produce straight-line graphs that lean left !( ) or right !( ) .

II) Linear equations can be graphed on a grid using only the x & y-intercepts. A) USE THESE STEPS TO GRAPH A LINEAR EQUATION USING THE x & y-INTERCEPTS

1: Calculate the y-intercept. 2: Calculate the x-intercept. 3: Set up a grid (REMEMBER TO LABLE AND SHOW THE SCALE OF EACH AXIS). 4: Plot the x & y-intercepts. 5: Use a ruler to connect the x & y-intercepts (REMEMBER TO LABLE THE LINE WITH THE EQUATION).



B) SAMPLE PROBLEMS 3: Graph these linear equations using the x & y-intercepts. Be sure you understand and memorize the process used to graph them. 1) y = 4x −7 2) 2x + 5y = 8

1: Calculate the y-intercept. (REMEMBER IT OCCURS WHEN x = 0)

2: Calculate the x-intercept. (REMEMBER IT OCCURS WHEN y = 0)

3: Set up a grid (REMEMBER TO LABLE AND SHOW THE SCALE OF EACH AXIS).

4: Plot the x & y-intercepts.

5: Use a ruler to connect the x & y-intercepts (REMEMBER TO LABLE THE LINE WITH THE EQUATION).

C) REQUIRED PRACTICE 3: Instructions: Graph these linear equations using the x & y-intercepts. (You can use the grids provided on pages 7, 8, 9 & 10 of these notes. {Ans. See your teacher.}

1) y = x +5 2) y = −2x + 4 3) 3x + 4y =12 4) −3x−5y =18

5) 2y = −3x + 4 6) y = −3x−5 7) 2x−3 4− y( )=1 8) 23 x + 3

2y = 6

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10



MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) SLOPE y-INTERCEPT FORM = a linear equation is in slope y-intercept form when it is written in this pattern:

y = mx +b where m ≠ 0 . Slope y-intercept form is also called general form.

2) SLOPE = m = a number describing the steepness of a graphed line. Slope is always written as a fraction.

3) y-INTERCEPT = b = the point where the graphed line touches or crosses the y-axis.

SLOPE y-INTERCEPT FORM I) WRITING A LINEAR EQUATION IN SLOPE y-INTERCEPT FORM

A) A LINEAR EQUATION IS IN SLOPE Y-INTERCEPT FORM WHEN IT IS WRITTEN IN THIS PATTERN: y = mx + b where m is the slope and b is the y-intercept. The slope, m, is a measure of how much the linear equation’s graph leans. The y-intercept is the point where the linear equation’s graph touches or crosses the y-axis of the grid. Study these examples carefully. Be sure you understand why the linear equations in the left hand column are in slope y-intercept form while the linear equations in the right hand column are not in slope y-intercept form.

SLOPE y-INTERCEPT FORM NOT SLOPE y-INTERCEPT FORM

y = x +1

y = −x −0.5

y = 45 x + 3

y = −12 x −18

y−x = 1

5y = 4x +15

y + 0.5 = −x

y = −2− 43 x

1)The linear equations in the left hand column are in slope y-intercept form because the y-term is isolated (by itself) and on the left side of the equal sign.

2) The linear equations in the right hand column are in NOT in slope y-intercept form. a) y−x = 1 is not in slope y-intercept form because the y-term is not isolated (by itself).

b) 5y = 4x +15 is not in slope y-intercept form because the y-term has a coefficient, it is not isolated.

c) y + 0.5 = −x is not in slope y-intercept form because the y-term is not isolated.

d) y = −2− 43 x is not in slope y-intercept form because the constant term is written before the x-term.

B) The process used to write a linear equation in slope y-intercept form is the same process used to isolate a variable described on page 2 of Topic 8. 1) SAMPLE PROBLEMS 7: Study these examples carefully. Be sure you understand and memorize the

process used to complete them. INSTRUCTIONS: Write these linear equations in slope y-intercept form. a) 3y+4x= 6 b) 3x + 43 y =

−4

2) REQUIRED PRACTICE 4: Write these equations in slope y-intercept form. SHOW YOUR PROCESS!! {Answers are on page 7 of these notes.} 1) 2x −y = 3 2)

−5x − 2

3y = 6 3) 32 y + 2x − 3 = 0



II) WHEN THE EQUATION OF A LINEAR EQUATION IS WRITTEN IN SLOPE y-INTERCEPT FORM: y =mx +b , two extremely valuable pieces of information are instantly revealed: The FIRST is b, which is the y-intercept; The SECOND is m, which is the slope. You must be able to state the y-intercept and the slope of any linear equation. A) SAMPLE PROBLEMS 5: Study these examples carefully. Be sure you understand and memorize the process

used to complete them. INSTRUCTIONS: Write the y-intercept and the slope of these linear equations. LINEAR EQUATION b m

1) y = 3x − 2

2) y = −58 x + 3

3) y = x

B) REQUIRED PRACTICE 5: Write the y-intercept and the slope of these linear equations. {Answers are on page 7 of these notes.} 1) y = 2x − 3 2) y = 32 x + 6 3) y = −9

4 x −7

EXPLAINING SLOPE y-INTERCEPT FORM

I) THE y-intercept AND THE SLOPE ARE ALL THAT YOU NEED TO DRAW THE GRAPH OF ANY LINEAR EQUATION A) The y-intercept, b, is the point where the graph touches or crosses the y-axis. Study this graph.

1) The graph to the right has the equation written in slope y-intercept form as y = 3

4 x +1 . This equation has a y-intercept of b = 1 which means its graph touches or crosses the y-axis at y = 1.

B) The SLOPE, m, is a number describing the steepness of a linear function’s graph. Slope is best written as the

fraction m = riserun

, which means m = up or down on the y−axisleft or right on the x −axis

. Slope is used to create points on a grid,

which are connected together to create the linear equation's leaning straight-line graph. 1) i.e. If

m = 4

3 , starting at the y-intercept, move positive 4 spaces up the y-axis, then move positive 3 spaces right on the x-axis, then plot a point. Continuing to use this pattern of up 4, right 3, plotting points until you run out of space on the grid.

2) i.e. If m = −25 , starting at the y-intercept, move negative 2 spaces down the y-axis, then move positive 5

spaces right on the x-axis, then plot a point. Continuing to use this pattern of down 2, right 5, plot a point until you run out of grid. You can also view this slope as m = −2

5 , which means starting at the y-intercept, more positive 2 spaces up on the y-axis, then move negative 5 spaces left on the x-axis, then plot a point. Continue using this pattern of up 2, left 5, plotting points until you run out of space on the grid.



GRAPHING LINEAR EQUATIONS USING SLOPE y-INTERCEPT FORM I) IN ORDER TO GRAPH LINEAR EQUATIONS, THEIR EQUATIONS MUST BE WRITTEN IN SLOPE y-INTERCEPT FORM,

WHICH IS y =mx +b , where m ≠ 0 . When a linear equation is written in slope y-intercept form, its graph can be drawn on a grid by one of these three methods.

1. Create a table of ordered pairs by choosing values of x, to substituting them into the equation, calculating values of y, then plotting the ordered pairs on a grid.

2. Input the equation into the graphing calculator, obtain the ordered pairs from the TABLE window of the graphing calculator, then plot the ordered pairs on a grid.

3. Use the y-intercept and the slope from the equation to draw the graph directly from the equation.

II) USE THESE STEPS TO GRAPH A LINEAR EQUATION

STEP 1: Write the linear function in slope y-intercept form. STEP 2: Set up a grid and plot the y-intercept. STEP 3: Use the slope to create as many other points as can fit on the grid. STEP 4: Use a RULER to connect the points; then label the graph with the equation.

A) SAMPLE PROBLEMS 6: Study these examples carefully. Be sure you understand and memorize the process used to complete them. 1) Graph y = 3x − 2

STEP 1: Write the linear equation in slope y-intercept form.

STEP 2: Set up a grid and plot the y-intercept. y-intercept =

STEP 3: Use the slope to create as many other points as can fit on the grid.

slope =

STEP 4: Connect points with a RULER then label the graph with the equation.

2) Graph 3y+4x= 9 STEP 1: Write the linear equation in slope y-

intercept form.

STEP 2: Set up a grid and plot the y-intercept.

y-intercept =

STEP 3: Use the slope to create as many other points as can fit on the grid.

slope =

STEP 4: Connect points with a RULER then label the graph with the equation.

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10



B) REQUIRED PRACTICE 6: Graph these linear equations on a grid. You can use the grids on pages 7, 8, 9 & 10. SHOW THE PROCESS!! {Ans. See your teacher} 1) y = 23 x − 3 2) y = −3

4 x + 5 3) y = x

4) 2x + 7y = 14 5) 3y + 5x = −12 6) 4y− 3x −16 = 0

ANSWERS TO THE REQUIRED PRACTICE

INVESTIGATION 1 from page 1 1) 0, 2( ); 0, 3( ) 2) x = 0 3) x = 0

Required Practice 1 from page 1 1) y = −6 2) y = −10 3) y = −6 4) y = 0 5) y = −3 6) y = 2 7) y = −13 8) y = 1

3 9) y = 0

INVESTIGATION 2 from page 2 1a)

−5, 0( ); 2, 0( ) 1b) y = 0 1c) y = 0

Required Practice 2 from page 2 1) x = 6 2) x = 2 3) x = 15

2

Required Practice 4 from page 4 1) y = 2x −3 2) y = −15

2 x −9 3) y = −43 x + 2

Required Practice 5 from page 5 1) b = −3, m = 2

1 2) b = 6, m = 32 3) b = −7, m = −9

4

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10



DO NOT USE FOR ASSIGNMENTS

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10




x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

fom 11-le3-graphing linear equations review-2...

Documents