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

Graphical Features of Exponential Functions

Graphically, in the role
f(ten) = ab ¹⁰.

a is the vertical intercept of the graph.
b determines the rate at which the graph grows:
- the part will increase if b > i,
- the function will subtract if 0 < b < ane.
The graph will have a horizontal asymptote at y = 0.
The graph volition exist concave up if a>0; concave downwardly if
a < 0.
The domain of the part 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 function, it can be helpful to remember that the graph volition pass through the points (0,
a) and (1, ab).

The value
b
will determine the function'due south long run behavior:

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

Example 6

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

This graph will accept a vertical intercept at (0,4), and pass through the point
[latex]\displaystyle{({1},frac{{four}}{{3}})}[/latex]. Since b < ane, the graph will be decreasing towards nix. Since a > 0, the graph will exist concave up.

graph

We can also come across from the graph the long run behavior: as
10 → ∞ , f(ten) → 0, and as x → –∞, f(ten) → ∞ .

To become a better feeling for the outcome of
a and b on the graph, examine the sets of graphs beneath. The first ready shows various graphs, where a remains the same and we only alter the value for b. Notice that the closer the value of is to 1, the less steep the graph volition be.

graph

Changing the value of
b
.

In the side by side set of graphs,
a
is altered and our value for
b remains the same.

graph

Irresolute the value of
a.

Find that changing the value for a changes the vertical intercept. Since
a
is multiplying the
b¹⁰ term, a acts equally a vertical stretch factor, not every bit a shift. Observe as well that the long run behavior for all of these functions is the aforementioned because the growth factor did not change and none of these values introduced a vertical flip.

Try it for yourself using
this applet.

Example 7

Match each equation with its graph.

[latex]\displaystyle{f{{({x})}}}={2}{({i.3})}^{{x}}[/latex]
[latex]\displaystyle{g{{({x})}}}={2}{({1.8})}^{{x}}[/latex]
[latex]\displaystyle{h}{({x})}={4}{({one.3})}^{{x}}[/latex]
[latex]\displaystyle{1000}{({x})}={iv}{({0.7})}^{{ten}}[/latex]

graph

The graph of
yard(x ) is the easiest to identify, since information technology is the only equation with a growth cistron less than 1, which will produce a decreasing graph. The graph of h(10 ) can be identified as the simply growing exponential part with a vertical intercept at (0,4). The graphs of f(x ) and m(x ) both have a vertical intercept at (0,two), but since 1000(x ) has a larger growth gene, nosotros can identify information technology every bit the graph increasing faster.