Match the graphs to the equations.
Graphs:
A

B

C

D

E

Equations:
(1) , (2) , (3) , (4) , (5)
Starting with the equations,
(1) is a linear equation (y = mx + c) so matches the only straight line, graph (D)
(2) is an exponential equation with a positive coefficient so matches graph (A)
(3) is a cubic equation with a negative coefficient so matches graph (E)
(4) is a reciprocal equation (notice that it takes the same form as inverse proportion) with a positive coefficient so matches graph (B)
(5) is a quadratic equation with a negative coefficient so matches graph (C)
Graph (A) → Equation (2)
Graph (B) → Equation (4)
Graph (C) → Equation (5)
Graph (D) → Equation (1)
Graph (E) → Equation (3)
A quadratic is a function of the form where is not zero
They are a very common type of function in mathematics, so it is important to know their key features
a)
Sketch the graph of showing the and intercepts
It is a positive quadratic, so will be a ushape
The at the end is the intercept, so this graph crosses the axis at (0,6)
Factorise
Solve
So the roots of the graph are
(2,0) and (3,0)
b)
Sketch the graph of showing the intercept and the turning point
It is a positive quadratic, so will be a ushape
The at the end is the intercept, so this graph crosses the yaxis at
(0,13)
We can find the minimum point (it will be a minimum as it is a positive quadratic) by completing the square:
This shows that the minimum point will be
(3,4)
As the minimum point is above the axis, this means the graph will not cross the axis i.e. it has no roots
We could also show that there are no roots by trying to solve
If we use the quadratic formula, we will find that is the square root of a negative number, which is not a real number, which means there are no real solutions, and hence no roots
Sketch the graph of showing the root(s), intercept, and turning point
It is a negative quadratic, so will be an nshape
The at the end is the intercept, so this graph crosses the axis at (0, 4)
We can find the maximum point (it will be a maximum as it is a negative quadratic) by completing the square:
This shows that the maximum point will be
(2, 0)
As the maximum is on the axis, there is only one root
We could also show that there is only one root by solving
If you use the quadratic formula, you will find that the two solutions for are the same number; in this case 2
(Press = when finished)
(If you are asked for another function, g(x), just press enter again)Calculator Allowed
Carefully plot the points from your table of values in (a) on the grid, noting the different scales on the and axes
For example, the first column represents the point
After plotting the points, join them with a smooth curve do not use a ruler!
It is best practice to label the curve with its equation