Sketching Graphs

Question 1

What are the key points when sketching a quadratic graph?

Answer 1

1) SHAPE
A quadratic is a smooth happy or sad face shaped curve.

2) ORIENTATION
Check if the curve should be in a positive or negative orientation.

3) INTERCEPTS 
Find any axis intercepts (let y = 0 for x-axis intercepts, and x = 0 for y-axis intercepts) and label them on the coordinate axes.

4) TOUCH OR CROSS?
Repeated roots are where the curve will touch the x-axis, and distinct roots are where the curve with cross the x-axis.

5) SYMMETRY
Ensure the curve is symmetrical - the turning point should appear to be half-way between any roots.

6) QUADRANT (NO REAL ROOTS)
If there are no x-axis intercepts, use completing the square or differentiation to find out which quadrant the turning point should be in.

Question 2

What are the key points when sketching a cubic graph?

Answer 2

1) SHAPE
A cubic is a smooth increasing or decreasing curve, with either a ‘wiggle’ or a ‘kink’ in the centre.

2) ORIENTATION
Check if the curve should be in a positive or negative orientation.

3) INTERCEPTS
Find any axis intercepts (let y = 0 for x-axis intercepts, and x = 0 for y-axis intercepts) and label them on the coordinate axes.

4) TOUCH or CROSS?
Repeated roots are where the curve will touch the x-axis, and distinct roots are where the curve with cross the x-axis.

5) QUADRANT (ONLY ONE ROOT)
If there is only one real root, use differentiation to determine where the turning points will be.

Question 3

What are the key points when sketching a basic exponential graph?

Answer 3

1) SHAPE
An exponential graph is a smooth curve that increases or decreases RAPIDLY.

2) INCREASING or DECREASING?
Check if a > 1, or if 0 < a < 1 .
If a > 1, the function will be increasing rapidly.
If 0 < a < 1 the function will be decreasing rapidly.

3) Y-INTERCEPT
Label the y-intercept, which will always be 1.

4) ASYMPTOTE
Indicate the asymptote clearly, which will always be the x-axis.

Question 4

What are the key points when sketching a basic logarithmic graph?

Answer 4

1) SHAPE
A logarithmic graph is a smooth curve that increases or decreases SLOWLY.

1) INCREASING or DECREASING?
Check if a > 1, or if 0 < a < 1 .
If a > 1, the function will be increasing slowly.
If 0 < a < 1 the function will be decreasing slowly.

2) X-INTERCEPT
Label the x-intercept, which will always be 1.

3) ASYMPTOTE
Indicate the asymptote clearly, which will always be the y-axis.

Question 5

What are the key points when sketching a basic reciprocal graph?

Answer 5

1) SHAPE
A reciprocal graph is a smooth ‘double’ curve that clings to the x and y axes in quadrants 1 and 3.

2) ASYMPTOTES
Indicate the asymptotes.
Both the x-axis and the y-axis will be asymptotes.

3) INTERCEPTS
There won’t be any!