Sec. 3 Graphs

Question 1

How do you find the equation of a line from two points?

Answer 1

1. find ‘m’, diff. in y/ diff in x
2. sub. this into the equ. y=mx+c and sub. the coords points into the equation
3. rearrange the equ. to find ‘c’
4. sub. back into y=mx+c

Question 2

What are the three methods for drawing straight ine graphs?
Explain these methods.

Answer 2

Table of values:

1. draw up a table with suitable values of x
2. Find the y values by sub. ing in each x value into the equation

Using the gradient:

1. Rearrange the equ. into the form y=mx+c
2. Put a dot on the y-axis for the value of c.
3. Using ‘m’ do up/down (numerator), left/right (denominator)

Sketching line graphs:

1. Set x=0 into the equ. and find y (where it crosses the y axis)
2. set y=0 into the equ. and find x (where it crosses the x axis)
3. Mark the points on the graph paper and draw a line though them.

Question 3

How do you find the midpoint of a line segment?

Answer 3

1. Add the x coords and divide by 2.
2. Do the same for the y coords.

Question 4

How can you use ratios to find the coordinates of a point on a line?

Answer 4

1. Find the difference between the coordinates of A and B (diff. in x coords and diff. in y coords).
2. look at the ratio you’ve been given which tell you what fraction along the line C is to the point, A.
3. Multiply the diff. in x and y coords by the fraction.
4. Add these to the coords of A to find C.

Question 5

How do you find the equation of a line parallel to another given line?

Answer 5

1. Find the gradient, ‘m’; it will be the same as the other line’s gradient as it is parallel.
2. Sub this gradient and the coordinate given (for x and y values) into the equ. y=mx+c
3. Solve to find ‘c’

Question 6

How do you find the line perpendicular to a given line, gien a coordinate?

Answer 6

1. find the gradient by changing the sign and finding the reciprocal.
2. sub. this into the equ. y=mx+c along with the x and y coordinate you’ve been given.
3. solve to find ‘c’
4. write the equ.

Question 7

How can you tell a graph is quadratic from its equ.?

Answer 7

It has an ‘x²’ in it.

Question 8

How do you plot/draw a quadratic graph?

Answer 8

1. Draw an appropriate table of values
2. sub. each x value into the given equ. to get each y value
3. Plot the points and draw an smooth curve - don’t count anomyles

Question 9

How do you sketch a quadratic graph?

Answer 9

1. Solve the equ. to find the x intercepts
2. Find the midpoint of the x intercepts (this is the turning point
3. sub this x value into the equ. to find the y coordinate.
4. Sketch

Question 10

How do you identify a cubic graph?

How do you draw a cubic graph?

Answer 10

It has an ‘x³’ term.

• Draw an appropriate table of values
• Sub. each x value into the given equation to get each y value.
• Plot the points and draw a smooth curve

Question 11

What do you need to remember about cubic graphs?

Answer 11

-x³ cubed graphs go down from th etp left.

+x³ graphs go up from bottom left

Question 12

What is the equation of a circle?

e.g. (x + 10)² + (y - 7)² =18

Answer 12

x2 + y2 = r2

centre: no.s in the brackets with the opposite signs as each coordinate. e.g. (-10,7)
radius: square root of the no. the equation is equal to.
e. g. =18 : 3√2

Question 13

How do you find the gradient to the tangent of a circle?

Answer 13

1. Find the gradient of the radius (diff. in y/ diff in x).
2. Find the perpendicular gradient
3. sub into y = mx + c
4. use the point where the tangent touches the circumference and sub. this into y = mx +c

Question 14

How can you show a coordinate lies on a circle?

e.g. Does (3,10) lie on the circle (x+3)² + (y-2)² radius:10

Answer 14

1. Sub. the coordinates into the equation
2. If it equals the radius squared it lies on the circle
3. e.g. (3,10)
4. (3+3) + (10-2)
5. 6² +8² =100

Question 15

What are the features of exponential graphs?

Answer 15

• Always above the x axis
• alwya go through the point (0,1)
• If k>1 the graph is +ve and the graph goes upwards
• If k is between 0 and 1, or the power is -ve, the graph is flipped horizontally

Question 16

Equations for reciprocal graph?

Answer 16

a) 1/x y=A/x xy=A

Question 17

b) What are the features of a reciprocal graph?

Answer 17

1. two inward pointing curves
2. -ve ones in opposite quadrants (For -ve, graph on top is on the LHS)
3. Both graphs don’t touch
4. Don’t exist for x=0
5. symmetrical about the lines y=x and y=-x

Question 18

What are the features of sine and cos graphs?

Answer 18

Question 19

What are the features of tanx graphs?

Answer 19

Question 20

a) How do you solve simultaneous equations using graphs?

Answer 20

1) Draw both graphs
2) Look for where the graphs cross

Question 21

a) How do you use a graph to estimate the solutions/roots to an equation (e.g. sinx=0.7) between to two limits (e.g. -180º and 180º)
b) Use the graph y=sinx to to estimate the solutions to sinx=0.7 between -180º and 180º

Answer 21

1) If the equation states sinx= (rather than y) draw a line across the point on the y axis where the equation equals to.

This is the graph of that equation.

2) Look at where this graph crosses a sin graph ( y=sinx)

Question 22

The graph of y=2x²-3x is shown below.

a) Use the graph to estimate both the roots of 2x²-3x=7
b) Find the equation of the line you would need to draw on the graph to solve 2x²-5x+1=0

Answer 22

Question 23

a) What is the format of the graph in graph transformations?
b) Where does the position of the variable tell you to move?
c) How are reflected grahs represented?

Answer 23

a) y = f(x)
b) Inside : left/right (y axis) , Outside : up/down (x axis)
c) -f(x) Reflect in x axis

f(-x) Reflect in y axis

Question 24

For y=f(x) find:

y = f(x) +2

y = f(x) -2

y = f(x+2)

Answer 24