Given F(X) = Ab-x C. How Would Increasing The Value Of A Change The Graph?

Graphical Features of Exponential Functions

Graphically, in the part
f(x) = ab ¹⁰.

a is the vertical intercept of the graph.
b determines the rate at which the graph grows:
- the office will increment if b > 1,
- the part will decrease if 0 < b < 1.
The graph will have a horizontal asymptote at y = 0.
The graph volition be concave upwardly if a>0; concave down if
a < 0.
The domain of the office is all real numbers.
The range of the function is (0,∞) if a > 0, and (−∞,0) if a < 0.

When sketching the graph of an exponential role, information technology can be helpful to think that the graph will pass through the points (0,
a) and (ane, ab).

The value
b
volition decide the role's long run beliefs:

If b > i, as x → ∞, f(x) → ∞, and as x → −∞, f(x) → 0.
If 0 < b < 1, equally x → ∞, f(10 ) → 0, and as ten → –∞, f (ten ) → ∞.

Example 6

Sketch a graph of
[latex]\displaystyle{f{{({10})}}}={4}{(\frac{{1}}{{3}})}^{{10}}[/latex]

This graph volition accept a vertical intercept at (0,four), and pass through the point
[latex]\displaystyle{({i},frac{{four}}{{three}})}[/latex]. Since b < 1, the graph will be decreasing towards goose egg. Since a > 0, the graph will be concave upward.

graph

We tin can besides see from the graph the long run behavior: equally
x → ∞ , f(x) → 0, and as x → –∞, f(x) → ∞ .

To get a ameliorate feeling for the effect of
a and b on the graph, examine the sets of graphs below. The beginning set shows various graphs, where a remains the same and nosotros only modify the value for b. Notice that the closer the value of is to i, the less steep the graph will exist.

graph

Changing the value of
b
.

In the side by side gear up of graphs,
a
is altered and our value for
b remains the aforementioned.

graph

Changing the value of
a.

Notice that changing the value for a changes the vertical intercept. Since
a
is multiplying the
b^x term, a acts as a vertical stretch cistron, not every bit a shift. Notice also that the long run beliefs for all of these functions is the aforementioned because the growth factor did not alter and none of these values introduced a vertical flip.

Try it for yourself using
this applet.

Case seven

Match each equation with its graph.

[latex]\displaystyle{f{{({x})}}}={ii}{({1.3})}^{{x}}[/latex]
[latex]\displaystyle{g{{({x})}}}={two}{({i.eight})}^{{x}}[/latex]
[latex]\displaystyle{h}{({ten})}={4}{({1.3})}^{{10}}[/latex]
[latex]\displaystyle{grand}{({x})}={4}{({0.7})}^{{10}}[/latex]

graph

The graph of
k(ten ) is the easiest to identify, since information technology is the only equation with a growth factor less than one, which volition produce a decreasing graph. The graph of h(x ) tin be identified as the but growing exponential function with a vertical intercept at (0,4). The graphs of f(x ) and g(ten ) both have a vertical intercept at (0,ii), but since g(x ) has a larger growth factor, we tin identify it as the graph increasing faster.