algebra graphs. plotting points - to draw straight line graphs we can use a rule to find and plot...
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![Page 1: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/1.jpg)
Algebra Algebra GraphsGraphs
![Page 2: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/2.jpg)
Plotting PointsPlotting Points- To draw straight line graphs we can use a rule to find and plot co-ordinatese.g. Complete the tables below to find co-ordinates in order to plot the following straight lines:a) y = 2x b) y = ½x – 1 c) y = -3x + 2 x y = 2x y = ½x –
1
-2
-1
0
1
2x y = -3x +
2
-2
-1
0
1
2
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
12345
2 x -1
2 x -2
0
2
-4
4
-2 ½ x -1 – 1
½ x -2 – 1
-1
-½
-2
0
-1 ½
-3 x -1 + 2
-3 x -2 + 2
2
-1
8
-4
5
![Page 3: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/3.jpg)
Gradients of LinesGradients of Lines- The gradient is a number that tells us how steep a line is.- The formula for gradient is:Gradient = rise
run
1st point
2nd point
rise
run
e.g. Write the gradients of lines A and B
A
B
A =
B =
4
6 8
4
4 = 18 2
6 = 34 2
e.g. Draw lines with the following gradientsa) 1 b) 3 c) 2 2 5
To draw, write gradients as fractions
= 3 1
When calculating gradients it is best to write as simplest fraction
![Page 4: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/4.jpg)
y = mxy = mx- This is a rule for a straight line, where the gradient (m) is the number directly in front of the x- When drawing graphs of the form y = mx, the line always goes through the origin i.e. (0,0)e.g. Draw the following lines:a) y = 2x b) y = 4x c) y = 3x 5 4
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
12345
1. Step off the gradient from the origin (0,0) 2. Join the plotted point back to the origin
= 4x 1
To draw, always write gradients as fractions
gradient
![Page 5: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/5.jpg)
Negative Negative GradientsGradients
e.g. Write the gradients of lines A and B
A =
B =
-3
2
10
-5
-5 = -110 2
-3 2
A
B
When calculating gradients it is best to write as simplest fraction
![Page 6: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/6.jpg)
e.g. Draw the following lines:a) y = -2x b) y = -4x c) y = -3x 5 4
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
12345
1. Step off the gradient from the origin (0,0) 2. Join the plotted point back to the origin
= -4x 1
To draw, always write gradients as fractions
gradient
![Page 7: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/7.jpg)
InterceptsIntercepts- Is a number telling us where a line crosses the y-axis (vertical axis)i.e. The line y = mx + c has m as the gradient and c as the intercept e.g. Write the intercepts of the lines A, B and C
x-10 -8 -6 -4 -2 2 4 6 8 10
y
-10
-8
-6
-4
-2
2
4
6
8
10
A
B
C
A =
B =
C =
8
4
-3
![Page 8: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/8.jpg)
Drawing Lines: Gradient and Intercept Drawing Lines: Gradient and Intercept MethodMethod- A straight line can be expressed using the rule y = mx + c
e.g. Draw the following lines:a) y = 1x + 2 b) y = -3x – 2 c) y = -4x + 8 2 7
x-10 -8 -6 -4 -2 2 4 6 8 10
y
-10
-8
-6
-4
-2
2
4
6
8
10To draw:1. Mark in intercept2. Step off gradient3. Join up points
= -3x – 2 1
Note: Any rule with no number in front of x has a gradient of 1 1e.g. y = x – 1
![Page 9: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/9.jpg)
Writing Equations of LinesWriting Equations of Lines- A straight line can be expressed using the rule y = mx + c
e.g. Write equations for the following lines
x-10 -8 -6 -4 -2 2 4 6 8 10
y
-10
-8
-6
-4
-2
2
4
6
8
10
A
B
C
A: B: C:m = c = m = c = m = c =
y = 3x – 6 4
y = -2x + 1 3
y = 4x + 4 1
34
-2 3
41 -6 +1 +4
![Page 10: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/10.jpg)
Horizontal and Vertical LinesHorizontal and Vertical Lines- Horizontal lines have a gradient of:
0Rule: y = c (c is the y-axis intercept)- Vertical lines have a
gradient that is:undefined
Rule: x = c (c is the x-axis intercept)e.g. Draw or write equations for the following lines:
a) y = 2 b) c) x = 4 d)
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
12345
x = -1y = -3
b)
d)
![Page 11: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/11.jpg)
Writing Equations Cont.Writing Equations Cont.When you are given two points and are expected to write an equation:- One method is set up a set of axes and plot the two points.
e.g. Write an equation for the line joining the points A=(1, 3) and B = (3, -1)
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
12345 m = -2
1c = 5
y = -2x + 5
Sometimes when plotting the points, you may need to extend the axes to find the intercept.
- Or, substitute the gradient and a point into y = mx + c to find ‘c’, the intercept
m = -2 1
using point
(1, 3)
y = mx + c 3 = -2 x 1 + c 3 = -2 + c 5 = c
+2 +2
y = -2x + 5
![Page 12: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/12.jpg)
Equations in the Form ‘ax + by = c’Equations in the Form ‘ax + by = c’- Can use the cover up rule to find the two intercepts:
e.g. Draw the following lines:a) 2y – x = 4 b) 4x – 3y =12
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5
-4
-3
-2
-1
1
2
3
4
51. Cover up ‘y’ term to find x intercept
- x = 4÷ -1 ÷ -1
x = -4
2. Cover up ‘x’ term to find y intercept
2y = 4÷ 2 ÷ 2
y = 2
3. Join up intercepts with a straight line
4x = 12÷ 4 ÷ 4
x = 3
-3y = 12÷ -3 ÷ -3
y = -4
It is also possible to rearrange equations into the form y = mx + ce.g. Rearrange 2x – y = 6
-2x -2x- y = 6 – 2x÷ -1 ÷ -1
y = -6 + 2xy = 2x – 6
![Page 13: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/13.jpg)
x1 2 3 4 5 6 7 8 9 10
y
102030405060708090
100110120130140
ApplicationsApplicationse.g. A pizzeria specializes in selling large size pizzas. The relationship between x, number of pizzas sold daily, and y, the daily costs is given by the equation, y = 10x + 50
1. Draw a graph of the equation
2. What are the costs if they sell 8 pizzas?$1303a. What is the cost per pizza?$103b. How is this shown by the graph?
The gradient of the line4a. What are the costs
if they sell no pizzas?
$504b. How is this shown by the graph?
Where the line crosses the y-axis
![Page 14: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/14.jpg)
The Basic The Basic ParabolaParabola
- The parabola is a quadratic graph linking y and x2- The basic parabola is y = x2
e.g. Complete the table below by using the rule y = x2 to find and plot co-ordinates to draw the basic parabola.
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-2-1
123456789
10
x y = x2
-2
-1
0
1
2
(-1)2
(-2)2
0
1
4
4
1
Note: the points of a basic parabola are easily drawn from the vertex by stepping out one and up one, then out two and up four, then out three and up nine etc...
VERTEX
![Page 15: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/15.jpg)
Plotting PointsPlotting Points- As with straight line graphs we can use a rule to find and plot co-ordinates in order to draw any parabola.
e.g. Complete the tables below to find co-ordinates in order to plot the following parabolas:a) y = x2 – 3 b) y = x2 + 2 c) y = (x + 1)2 d) y = (x – 1)2 x y = x2 – 3 y = x2 + 2
-2
-1
0
1
2
x y = (x + 1)2
y = (x – 1)2
-2
-1
0
1
2
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-4-3-2-1
12345678
(-1)2 – 3
(-2)2 – 3
-3
-2
1
1
-2 (-1)2 + 2
(-2)2 + 2
2
3
6
6
3
(-1 + 1)2
(-2 + 1)2
1
4
1
9
0 (-1 – 1)2
(-2 – 1)2
1
0
9
1
4
![Page 16: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/16.jpg)
Transformations of the Basic Transformations of the Basic ParabolaParabola
1. Up or Down Movement- When a number is added or subtracted at the end, the basic parabola moves vertically
e.g. Draw the following parabolas:a) y = x2 b) y = x2 + 1 c) y = x2 – 5
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
1234567
To draw vertical transformations, first find the position of the vertex Then draw in basic parabola shape
![Page 17: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/17.jpg)
2. Left or Right Movement
- When a number is added or subtracted in the brackets, the basic parabola moves horizontally but opposite in direction
e.g. Draw the following parabolas:a) y = x2 b) y = (x + 3)2 c) y = (x – 2)2
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
1234567
To draw horizontal transformations, first find the position of the vertex Then draw in basic parabola shape
![Page 18: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/18.jpg)
3. Combined Movements
e.g. Draw the following parabolas:a) y = (x – 4)2 – 8 b) y = (x + 3)2 + 3 c) y = (x – 7)2 + 4 d) y = (x + 6)2 – 5
x-10 -8 -6 -4 -2 2 4 6 8 10
y
-10
-8
-6
-4
-2
2
4
6
8
10
To draw combined transformations, first find the position of the vertex Then draw in basic parabola shape
![Page 19: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/19.jpg)
Changing the Shape of the Basic Changing the Shape of the Basic ParabolaParabola
1. When x2 is multiplied by a positive number other than 1- the parabola becomes wider or narrower- Set up a table and use the rule to find and plot co-ordinates
e.g. Complete the tables and draw y = 2x2 and y = ¼x2
x y = 2x2 y = ¼x2
-2
-1
0
1
2
2 × (-1)2
2 × (-2)2
0
2
8
8
2
¼ × (-2)2
¼ × (-1)2
1
¼
0
¼ 1
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-2-1
123456789
10
Use the grid to determine the x-values to put into your table
![Page 20: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/20.jpg)
1. When x2 is multiplied by a negative number- it produces an upside down parabola- all transformations are the same as for a regular parabola
e.g. Draw the following parabolas: y = -x2
y = -(x + 2)2
y = -(x – 1)2 + 2 First find placement of the vertex
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
12345
When plotting points move down instead of up.
![Page 21: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/21.jpg)
Factorised Factorised ParabolasParabolasMethod 1: Set up a table, calculate and plot points
e.g. Draw the parabola y = (x – 3)(x + 1)Use the grid to determine the x-values to put into your table
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
12345
x y = (x – 3)(x + 1)
-3
-2
-1
0
1
2
3
(-3 – 3)(-3 + 1)12
(-2 – 3)(-2 + 1)5
0-3
-4
-3
0
![Page 22: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/22.jpg)
Method 2: Calculating and plotting specific featurese.g. Draw the parabola y = (x – 3)(x + 1)
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
12345
1. x-axis intercepts (where y = 0) solving quadratics: 0 = (x – 3)(x + 1)x = 3 and -1
2. y-axis intercept (where x = 0) y = (0 – 3)(0 + 1)y = -3
3. The position of the vertex- is halfway between x-axis intercepts- substitute x co-ordinate into equation to find y co-ordinate
y = (1 – 3)(1 + 1)y = -4
4. Join the points with a smooth curve
Vertex = (1, -4)
![Page 23: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/23.jpg)
e.g. Draw the parabola y = x(x – 4)
x-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8
y
-9-8-7-6-5-4-3-2-1
12345678
1. x-axis intercepts 0 = x(x – 4) x = 0 and 4 2. y-axis intercept y = 0(0 – 4) y = 0 3. Position of vertex y = 2(2 – 4) y = -4 Vertex = (2, -4)
e.g. Draw the parabola y = (1 – x)(x – 5)1. x-axis intercepts 2. y-axis intercept 3. Position of vertex
0 = (1 – x)(x – 5) x = 1 and 5 y = (1 – 0)(0 – 5) y = -5 y = (1 – 3)(3 – 5) y = 4 Vertex = (3, 4)
Note: -x indicates parabola will be upside down
![Page 24: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/24.jpg)
Expanded Form Expanded Form ParabolasParabolas
- Remember you can always set up a table and calculate co-ordinates to plot.- Or simply factorise the expression and plot specific points as shown earlier
e.g. Draw the parabolas y = x2 – 2x – 8 and y = x2 + 2x
Factorised Expression y = x(x + 2)y = (x – 4)(x + 2)
1. x-axis intercepts 2. y-axis intercept 3. Position of vertex
x = -2 and 4 y = -8 Vertex = (1, -9)
x = 0 and -2 y = 0 Vertex = (-1, -1)
x-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8
-10-9-8-7-6-5-4-3-2-1
12345678
![Page 25: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/25.jpg)
Writing EquationsWriting Equations- If the parabola intercepts x-axis, you can substitute into y = (x – a)(x – b) - Or, you can substitute the vertex co-ordinates into y = (x – a)2 + b
e.g. Write equations for the following parabolas
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
12345
(a)
(b)
(c)
a)
c)
b)
Always substitute in the opposite sign x-value
y = (x – 2)(x – 4)or
Vertex = (3, -1)y = (x – 3)2 – 1
Vertex = (-2, 1)y = (x + 2)2 + 1
y = (x + 1)(x + 5)or
Vertex = (-3, 4)y = (x + 3)2 + 4
Add in a negative sign if parabola upside down
-
-
![Page 26: Algebra Graphs. Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.g. Complete the tables below to find](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649dc35503460f94ab635c/html5/thumbnails/26.jpg)
Writing Harder Writing Harder EquationsEquations
- Used when the co-efficient is not equal to 1. - Use either of the equations y = k(x – a)(x – b) or y = k(x + c)2 + d
e.g. Write equations for the following parabolas
x-4 -2 2 4 6 8 10
y
-4-2
2468
1012a) b)
x5 10 15 20 25 30 35
y
10
20
30
40
50
y = k(x – 1)(x – 5)
Substitute in the values of a specific point to find the coefficient k
-2 = k(3 – 1)(3 – 5)-2 = k×-4
0.5 = k
y = 0.5(x – 1)(x – 5)
y = k(x – 20)2 + 40
0 = k(10 – 20)2 + 40 0 = k×100 + 40
-40 = k×100
y = -0.4(x – 20)2 + 40-0.4 = k
Coefficients can be written as decimals or fractions
(3, -2) (10, 0)