a The function of a(x) is exponential and b(x) is a quadratic. Use the information in the table and the diagram to write their equations.
B
b An exponential function will always overtake a power function when x gets sufficiently large.
A
a See solution.
B
bDiagram:
Explanation: The graph of a(x) will at some point intersect b(x) and then exceed it.
Practice makes perfect
a Let's start with a(x). There's nothing that indicates that this function has the shape of a parabola or any other polynomial function of a higher degree. Therefore, we will assume it is either linear or exponential.
Linear:& y=mx+b
Exponential:& y=ab^x
Examining the table's three first data points, we notice that it cannot be linear as this would imply a constant rate of change, which is not the case here. If it is an exponential function, there is a common ratio between consecutive data points. Let's investigate this.
As we can see, we have a common ratio between consecutive data points which means this is in fact an exponential function with an initial value of a=10 and a common ratio of b=1.05. With this information, we can write the function.
a(x)= 10(1.05)^x
Next we will have a look at b(x). From the graph, we notice that it has the shape of a parabola which means this is a quadratic function. We also see that it has its vertex in (-2,24) and a y-intercept of (0,24). When we know the vertex of a parabola we can write it in graphing form.
Graphing Form:& f(x)=a(x-h)^2+k
Vertex:& (h,k)
Let's substitute h=-2 and k=4 into the graphing form of a quadratic.
b(x)=a(x-(-2))^2+4
⇕
b(x)=a(x+2)^2+4
To find the value of a we will substitute the y-intercept into the function and then proceed to isolate a by performing inverse operations.
