Sec 3 Graphs

Question 1

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

Answer 1

1- Use diff. in y/ diff. in x, to find the gradient of the line

2- Sub. this into the equation y=mx+c

3- Sub. the points into the equation

4-rearrange the equation to find ‘c’

5- Sub back into y=mx+c

Question 2

Find the equation of the line that passes through the points (-2,9) and (3,-1)

Answer 2

Question 3

a) What are the three methods for Drawing straight line graphs?
b) Explain these methods

Answer 3

Table of values:

1- Draw up a table with suitable values of x

2-Find the y values by sub.ing each x value into the equation

Using y=mx=c:

1- Rearrange the equation into the form y=mx=c

2- Put a dot on the y axis for the value of c

3-Using ‘m’ go up/down and left/right the correct no of units. e.g. If ‘m’ is +1/2, Go 2 right, 1 up.

4-When you have several dots draw a line through them.

5-Check gradient looks right

Skething Line Graphs:

1- Set x=0 in the equation , and find y ( where it crosses the y axis)

2- Set y=0 in the equation, and find x (where it crosses the x axis)

3- Mark the points on the line and draw a line through them.

Question 4

Draw a graph of y= -2x+4 using the table of values method

Answer 4

Question 5

Draw the graph of 4y-2x= -4 using the y=mx+c method

Answer 5

Question 6

Sketch the straight line graph of y=3x-5

Answer 6

Question 7

How do you find the midpoint of a line segment?

Question 8

Points A and b have the coordinates (7,4) and (-1, -2) respectively.Find the midpoint.

Answer 8

[(7+-1) /2,(4+-2) /2]

(6/2,2/2)

=(3,1)

Question 9

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

Answer 9

1- Find the difference between the coordinates of A and B (diff. in x coord.s and diff. i y coord.s).

2-Look at the ratio you’ve been given which will tell you what fraction along the line A is to the point, C.

3- Multiply the diff in x and y coord.s by the fraction.

$- Add these to the coord.s of A to find C.

Question 10

Point a has the coordinates (-3, 5) and Point B has coord.s (18,33). Point C lies on the line segment AB, so that AC:CB= 4:3.

Find the coord. of C.

Answer 10

Question 11

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

Answer 11

1- Find the gradient; it will be same as the other line’s gradient as oi is parallel

2- Sub this gradient and the coordinate given (for x and y values) into the eqation y=mx+c

3- Solve to find ‘c’

Question 12

How do you find the perpendicular line of a given line given a coordinate.

Answer 12

1- Find the perpendicular gradient by changing the sign and finding the recipricol.

2- Sub this into the equation y=mx+c along with the x and coordinate you’ve been given.

3- Solve to find ‘c’

4-Write the equation

Question 13

1) How can you tell a graph is quadratic from its equation?
2) How do you plot a Quadratic graph?

Answer 13

1) It has an x^2 in it.

2)

• Draw an appropriate table of values (e.g.-4 to 4).
• Sub. each x value into the given equation to get each y value.
• Plot the points and draw a smooth curve.
• Dont count anomyles

Question 14

How do you sketch a quadratic graph?

Answer 14

1- Solve the equation to find the x-intercepts

2-

• To find the turning point of the curve find halfway between the x intercepts. (add them together and /by two)
• Sub this x value into the equation.
• The x coord is the one you sub.ed into the equation and the y coord is the no. you get from solving the equation.

3- Sketch

Question 15

Skecth the graph of -x²-2x+8=0

Answer 15

Question 16

a) How do you identify a cubic graph
b) how do you draw a cubic graph
3) what must you remember about cubic graphs

Answer 16

a) it has a value of x^3

b)

• Draw an appropriate table of values (e.g.-4 to 4).
• Sub. each x value into the given equation to get each y value.
• Plot the points and draw a smooth curve.

Question 17

a) What is the equation of a circle and what does this format describe?
b) How do you find the centre/ radius of a circle?
c) e.g. (x+10)²+(y-7)²=18

Answer 17

a) x²+y²=r²

b)Centre (-10,7) r=3./2

Question 18

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

Answer 18

a)

1-Find the gradient of the radius 9 (diff. in y/ diff. in x)

2-Find the perpendicular gadient

3-sub. into y=mx+c

4-Us ethe point where the tangent touches the circumference and sub. this into y=mx+c

Question 19

Find the equation of the tangent of the circle with the centre (0,0). the tangent touches the circle at the point (3,9)

Answer 19

a) find the gradient of the tangent
(diff. in y/ diff. in x) (0,0) (3,9)

9-0/3-0 = 9/3 =3

b) find the perp. m -1/m

-1/3

c) sub into y=mx+c

y=-1/3x+c

d) use the point where the tangent touches the circumference and sub. this into the equation

9=(1/3x×3)+c (3,9)

9=-1+c

c=10

y=-1/3x+10

Question 20

a) How can you show a coordinate lies in a circle?
e. g. Does (3,10) lie on the circle (x+3)²=(y-2)²

Answer 20

a)

• Sub. the coordinates into the equation
• If it equals the radius squared it lies on the circle
  e. g.

(3,10)

(3+3) = (10-2)

6²+8²=100

Question 21

What are the features of exponential graphs?

e.g.

The graph shows how the number of victims of an alien virus (N) increases in a science fiction film. The equation of the graph is N=fg^t

Where t is the no. of days in the film and f and g are +ive constants. Find the values of f and g.

Answer 21

Question 22

a) Equations for reciprocal
b) What are the features of a reciprocal graph?

Answer 22

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

Question 23

What are the features of sine and cos gaphs?

Answer 23

Question 24

What are the features of tanx graphs?

Answer 24

Question 25

a) How do you solve simultaneous equations using graphs? E.g. Solve the simultaneous equations x²⁺y²=16 and y=2x+1

Answer 25

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

Question 26

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 26

1) If the equation states sin**x**= (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 27

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 27

Question 28

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 28

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 29

For y=f(x) find: y = f(x) +2 y = f(x) -2 y = f(x+2)

Answer 29

Question 30

For y=f(x), find: y = -f(x)+2 y = f(-x)-1 y = f(x-3)-1

Answer 30

Question 31

Describe how a graph of y = f(x) is transformed into y = -f(-x)

Answer 31

Graph is reflected in the x axis and the y axis