Exponential functions are powerful tools for modeling growth and decay. They have a unique shape, starting slow and then rapidly increasing or decreasing. Understanding their key features and transformations is crucial for interpreting real-world scenarios.
Comparing exponential graphs reveals how different growth rates affect outcomes over time. This knowledge is essential for analyzing population dynamics, financial investments, and other phenomena that exhibit exponential behavior.
Exponential Function Graphs
Key features of exponential graphs
Exponential functions have the general form f(x)=abx where a represents the initial value (y-intercept) and b represents the base (growth or decay factor)
If b>1, the function exhibits (population growth)
If 0<b<1, the function exhibits (radioactive decay)
If b=e (Euler's number), the function is called a natural exponential function
y-intercept: the point where the graph intersects the y-axis, occurring at x=0
For f(x)=abx, the y-intercept is located at (0,a)
Horizontal asymptote: the line that the graph approaches as x approaches positive or negative infinity
For exponential growth, the horizontal asymptote is y=0 as x approaches negative infinity
For exponential decay, the horizontal asymptote is y=0 as x approaches positive infinity
End behavior: the graph's behavior as x approaches positive or negative infinity
For exponential growth, as x approaches positive infinity, f(x) approaches positive infinity
For exponential decay, as x approaches negative infinity, f(x) approaches positive infinity
Transformations of exponential functions
Shifts: translate the graph vertically or horizontally without altering its shape
: f(x)=abx+k translates the graph k units up if k>0 (upward shift) or down if k<0 (downward shift)
: f(x)=abx−h translates the graph h units right if h>0 (right shift) or left if h<0 (left shift)
Reflections: flip the graph across the x-axis or y-axis
across the x-axis: f(x)=−abx reflects the graph over the x-axis (upside-down)
Reflection across the y-axis: f(x)=ab−x reflects the graph over the y-axis (left-right)
Stretches and compressions: alter the graph's steepness or flatness
Vertical /: f(x)=cabx stretches the graph vertically by a factor of c if ∣c∣>1 (taller) or compresses it if 0<∣c∣<1 (flatter)
Horizontal stretch/compression: f(x)=abcx compresses the graph horizontally by a factor of c if ∣c∣>1 (narrower) or stretches it if 0<∣c∣<1 (wider)
Comparison of exponential graphs
Rate of change: the speed at which the function's output values change as the input values change
Exponential functions have a constant ratio between consecutive output values for equal increments of the input (constant percent change)
The larger the base b, the faster the rate of change (steeper graph)
Comparing exponential function graphs
Functions with the same base will never intersect, as they share the same rate of change (parallel graphs)
Functions with different bases may intersect at one point (point of intersection), but will diverge as x increases or decreases
The function with the larger base will grow or decay more rapidly than the function with the smaller base (steeper vs flatter graph)
Properties of exponential functions
Domain: All real numbers (continuous function)
Range: All positive real numbers (y > 0)
Exponential functions are monotonic functions, meaning they are either always increasing or always decreasing
Key Terms to Review (8)
Compression: Compression in algebra and trigonometry refers to the transformation that scales a graph towards an axis, making it narrower. This is achieved by multiplying the function by a constant factor.
Exponential decay: Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. Mathematically, it is expressed as $N(t) = N_0 e^{-kt}$, where $N_0$ is the initial quantity, $k$ is the decay constant, and $t$ is time.
Exponential growth: Exponential growth describes a process where the quantity increases at a rate proportional to its current value. This results in the quantity growing faster and faster over time.
Horizontal shift: A horizontal shift is a transformation that moves a graph left or right along the x-axis without changing its shape. It is represented by modifying the function as $f(x) \rightarrow f(x - h)$, where $h$ is the number of units shifted.
One-to-one: A one-to-one function is a function where each input corresponds to exactly one unique output, and each output corresponds to exactly one unique input. This ensures that no two different inputs produce the same output.
Reflection: Reflection is a transformation that flips a graph over a specific line, such as the x-axis or the y-axis, creating a mirror image. In algebra and trigonometry, it alters the sign of either the x-coordinates or y-coordinates of points on a graph.
Stretch: A stretch is a transformation that alters the shape of a graph by scaling it vertically or horizontally. It affects the distance between points on the graph without changing their relative positions.
Vertical shift: A vertical shift is a transformation that moves a graph up or down by adding or subtracting a constant to the function's output. This does not change the shape of the graph, only its position along the y-axis.