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Topic: Introduction to graphing: Plotting Points Name:________________________ Per:________________________ Date:________________________ Class:________________________ Notes: (Day 1) Label Quadrant _____ Quadrants Quadrant _____ ( ____ , ____ ) ( ____ , ____ ) here: ( ____ , ____ ) ___________ or ( ___ , ___ ) ( ____ , ____ ) ( ____ , ____ ) Quadrant _____ Quadrant _____ ( ____ , ____ ) Points Every point has ____ numbers in it The first number represents the __ value, the second the ___ value. You always start from the ___________ when plotting points. The x-value is how much you move ________ or ________ The y-value is how much you move _____ or _________ ( ____________ , ____________ ) ( __________ or _________ ) Practice ( __________ or _________ ) What quadrant is each point in? A (2,5) B ( -5 , 3 ) C ( -2 , -1 ) D ( -1 , 0 ) 1 Practice Plot the following points on the graph below. Label each point with its designated letter. plotting A (1,4) Practice Give the coordinates of each point. identifying points B(6,3) C ( -4 , 2 ) D ( -1 , -6 ) E ( 7 , -8 ) A ( ____ , ____ ) B ( ____ , ____ ) C ( ____ , ____ ) D ( ____ , ____ ) E ( ____ , ____ ) F ( ____ , ____ ) G ( ____ , ____ ) H ( ____ , ____ ) 2 Domain The ___ values. Range The ___ values. What is the domain of the function shown below? STAR released question #79 Domain: { ___ , ___ , ___ , ___ } Which line goes through the point (-2,-4)? D C B A Summary: The points (–4 , 3 ) and ( 3 , –4 ) appear to be similar, but are two unique points. Discuss the differences of the two points. 3 Topic: Graphing Lines: Slope-intercept form (Day 2) Name:________________________ Per:________________________ Date:________________________ Class:________________________ Notes: Slope-intercept form of a y = _______ + _____ line y-intercept The y-intercept is represented by the letter _____ . The y-intercept is where the line ________ the __-axis. It is also where you __________ each graph. slope The slope is represented by the letter ____. The slope is . You go ____ or ________, then to the __________. Positive slopes always go _____ and to the _________. They look like this. Positive slopes Negative slopes always go _______ and to the _________. They look like this. Negative slopes 4 Graphing a line 1) Always start at the ____________________. 2) From there graph the _____________. Practice A) Graph y 2 x4 3 1 B) Graph y x 5 2 C) Graph y 2 x6 1 5 D) Graph y 3 x 1 E) Graph y x 2 F) Graph y 2 x 6 G) Graph y 7 x4 8 What is the y-intercept of 4 x 2 y 12 ? STAR released question #23 How can we get the equation into slope-intercept form? _____________ Teacher guided question 4 x 2 y 12 If Then, 2 y = _______________ And, y = _______________ and our y-intercept is ______ Summary: Given the graph below, write the equation of the line. y 5 4 3 2 1 –5 –4 –3 –2 –1 –1 1 2 3 4 5 x Equation: ____________________________ –2 –3 –4 –5 Hint: What is the slope? Hint: What is the y-intercept? 7 Topic: Identifying equations of lines and slope (Day 3) Name:________________________ Per:________________________ Date:________________________ Class:________________________ Notes: Review Positive slopes go ___ and to the _________. Negative slopes go _______ and to the _________. Positive y-intercepts are above the ___________. Negative y-intercepts are below the ___________. Similar to STAR released What linear equation is shown on the graph below? question #24 Concept review 1 x 1 2 A y B 1 y x 1 2 C y D 1 y x 1 2 1 x 1 2 When graphing a line, START with the __________________, and then use the _____________ after that. To find the slope, find two ____________ where the line goes through. Count how far ______________ and to the _________ the line goes between those two points. ____________ too. 8 Slope concept The slope is represented by the letter ____ The slope is You go ____ or ________, then to the __________ Slope formula To find the slope, you are talking about finding the As a formula, this looks like Where do the 1s and 2s When you are given two ordered pairs (two points), in each come from? ordered pair you can label the __ and __. But since you have two different __s, and two different __s, you label the _________ ordered pair with ones, and the second ordered pair with ____. 2,1 3,9 Try it Label: Now find the slope Use the formula to find the slope of the line passing through 2,1 and and 3,9 . Simplify your answer if possible. Try again Find the slope passing through the points 4, 2 and 1, 6 Harder examples A) 1, 4 and 2, 10 9 B) 14, 25 and 37,60 C) STAR released question 8, 12 and 14, 20 Which best represents the graph of y 2 x 2 ? #25 A C B D 10 STAR released question #27 Which equation best represents the graph above? A yx Think it through B y 2x C y x2 D y 2x 2 Every line in the form of y = mx+b has ____ y-intercept. Every line in the form of y = mx+b has ____ x-intercept. Can a line pass through the x-axis twice? Why? Sometimes/always/never The graph of a linear function in the form y mx b has one xintercept. _______________________ 11 STAR released question Which equation represents the line shown in the graph below? #31 A y 2 x4 3 B y 2 x6 3 C y 3 x4 2 D y 3 x6 2 STAR based skill What is the slope of the line y STAR based skill What is the y-intercept of the equation y 2 x 3 ? A 0, 2 B 0, 2 C 0,3 D 0, 3 4 x7 ? 5 ______________ 12 Review The equation y 3 x 3 crosses the __ -axis at ____ and then goes _____ and to the _______. Summary: Explain how you determine which of the four numbers goes in each position of the formula when finding the slope. 13 Topic: Graphing linear equations: Standard form (Day 4) Name:________________________ Per:________________________ Date:________________________ Class:________________________ Notes: Review Slope-intercept form is ___ = _____ + _____. To be in slope-intercept form, _____ always has to be by itself! Standard form If an equation is given in standard form, the ___s and ____s are on the ______ side of the equation. It looks like: OR What to do OR Usually the best thing to do when an equation is given in standard form is to ____________ it to ________________ form. Practice Convert 2 x 3 y 12 to slope-intercept form. Are there any like terms? ____________________________ How many terms should this have in slope-intercept form? ________ What is the first thing to do? ________________________________ Show your work here: Try it on your own Convert 3x 4 y 24 to slope-intercept form. 14 Thinking it through Look at the problem at the bottom of this page. What equation is given? _____________________________ What will you need to do to determine the correct graph? ___________ _________________________________________________________ _________________________________________________________ Do that here: What do graphs A & C have in common? _______________________ What do graphs A & B have in common? _______________________ STAR based skill Which graph represents 5 x 6 y 12 ? y –5 –5 –4 –4 –3 –3 –2 –2 y 5 5 4 4 3 3 2 2 1 1 –1 –1 1 2 3 4 5 x –5 –4 –3 –2 –1 –1 –2 –2 –3 –3 –4 –4 –5 –5 A C y y 5 5 4 4 3 3 2 2 1 1 –1 –1 1 2 3 4 5 x –5 –4 –3 –2 –1 –1 –2 –2 –3 –3 –4 –4 –5 –5 B 1 2 3 4 5 x 1 2 3 4 5 x D 15 Practice converting 1) 4 x y 2 2) 4 x y 2 3) 2 x 3 y 1 4) 2 x 3 y 1 5) x y 4 6) x y 4 to slope-intercept form Summary: Sally says that 2 x y 4 in slope-intercept form is y 2 x 4 . Explain why this is not correct. 16 Topic: Graphing Linear Equations: Special Cases (Day 5) Name:________________________ Per:________________________ Date:________________________ Class:________________________ Notes: 2 unique cases Not every line has a y-intercept and a slope. A ___________________ line goes from ______ to ________ A ______________________ lines goes ____ and _________ Vertical lines Vertical lines pass through the ___ - axis, thus their equation looks like x = a, where a is a number. For example, x 3 is a vertical line. Example Look at the line passing through the points (4,5) and (4,0) How much does the line rise between thos two points? ______________ run? ________ So its slope is , which is ___________________. Thus we call the slope of a vertical line _________________. Please don’t call this ______ __________! Graph 1) x 2 What does x always If we have the line x 7 , the graph goes _____ and ________, equal in a vertical line? and every x value is equal to _______. So, no matter what y is, x will always be _____. 17 Horizontal lines Horizontal lines pass through the ____ - axis, thus their equation looks like y = b. For example, y 1 is a horizontal line. Example Look at the line passing through the points (0,5) and (7,5) How much does the line rise? ______________ run? ________ So its slope is , which is _____. Thus a horizontal line has a slope of ________________. Please don’t call this ____ ____________! Graph 1) y 4 What does y always If we have the line y 6 , the graph goes _____ and ________, equal in a horizontal line? and every y value is equal to _______. So, no matter what x is, y will always be _____. Summary: How can you tell whether a line will be horizontal or vertical? 18 Topic: Another way of graphing: Pointslope form (Day 6) Name:________________________ Per:________________________ Date:________________________ Class:________________________ Notes: Another form of a line y mx b is not the only way to graph a line. Instead of always having to start with the ___- intercept, there is another way to write an equation of a line that can start from any point on the graph. It is called __________ - __________ form. Point-slope form For any point x, y , and slope m, an equation of the line passing through the point with the slope can be written: How to use point-slope Instead of always having to start at the ___-_____________ and form then graphing the slope, you could start graphing at any ________ that the graph goes through, and then graph the ________ from there! When plugging the point 2,3 into the equation, the signs always _____________. For example, the line that passes through 2,3 with a slope of 4 looks like ____________________________ Try this 1) 3,5 , m = -2 Graph Graph the line that passes 2) 2, 4 , m = 1 through 2,3 with a slope of 1 . 2 19 STAR released question Which point lies on the line defined by 3x 6 y 2 ? 28 #28 A 0, 2 B 0, 6 1 C 1, 6 1 D 1, 3 List two ways that you could determine if a point is on a line. 1) _______________________________________________________ 2) _______________________________________________________ Attempt #1: ______________ the ___________ into the ____________. A 0, 2 B 0, 6 1 C 1, 6 1 D 1, 3 Attempt #2: ________________ the line and ____ which ________ the Hint: Begin by getting the line _____________ through. equation into _______________________ form. 3x 6 y 2 20 Try this Write the equation of a line that has a slope of -3 and passes through the point 2, 1 . STAR released question #29 STAR based question What is the equation for the line passing through the points graphed below? y 6 5 4 3 2 1 –6 –5 –4 –3 –2 –1 –1 1 2 3 4 5 6 x –2 –3 –4 –5 –6 STAR based question What is an equation of the line that passes through the points ? 21 Summary: Pat is given the points 2, 3 and 1,6 . Explain how Pat finds the equation of the line is y 3x 3 . 22 Topic: Another way of graphing: Coverup Method (Optional) Name:________________________ Per:________________________ Date:________________________ Class:________________________ Notes: Standard form When equations are given in standard form, sometimes you can graph the equation by finding the __ and __ intercepts. Note: This does not help you graph _________________, and it doesn’t always help you find the solution to a ______________. Intercepts When a point is on the x-axis (an x-intercept), the coordinates always look like ( __ , __ ). When a point is on the y-axis (an y-intercept), the coordinates always look like ( __ , __ ). This means That when you solve for an x-intercept, you plug in ___ for y, which means it _____________, and you can _________ it up. So when you solve for the y-intercept, you _________ up the x. Try this Graph 4 x 2 y 12 Find the intercepts only 1) 2 x y 4 2) 1 x 3y 6 2 23