Graphs 1
The graph below represents the following functions:
f : x -> ax2+bx+c and
g : x -> mx + k
1. Find the equation of f.
2. Determine the value of k.
3. What is the range of f?
4. For what value(s) of x is f–g = 0?
5. Determine the value(s) of x for which f < 0
SOLUTION
1.Find the equation of f.
Since we have 2 roots given and one other point, we will use this form of the equation:
y = a(x-x1)(x-x2)
Hence
EQUATION 1
y = a(x+3)(x-1)
Now, substitute the coordinate (2;10) to solve for a.
(Substitute x = 2 and y = 10)
10 = a(2+3)(2-1)
10 = a(5)(1)
10=5a
a=2
Now that we know the value of a, we substitute it back in EQUATION 1.
y = a(x+3)(x-1)
y=2(x+3)(x-1)
y= 2(x2+2x-3)
y=2x2+4x-6
2. Determine the value of k.
k is the y-intercept of the straight line. So we must work out the equation of the straight line first!
y = mx + c
y= 2x + c
Substitute (2;10) since this point lies on the straight line as well:
10=2(2) + c
10 = 4 + c
c = 6
3. What is the range of f?
This requires you to find out, the y-values that belong to the graph. We must find the turning point to do this.
Axis of symmetry:
y=2x2+4x-6
= 2(-1)2+4(-1)-6
=-8
Since this is a minimum value graph, the range is:
4. For what value(s) of x is f–g = 0?
In other words when is f = g. (where do the graphs intersect?)
At x = -3 and x = 2
5. Determine the value(s) of x for which f < 0Here we need to find out which part of the graph is below the x-axis.
Therefore our solution is:
