Problem 29 - Intermediate Algebra

Question

Graph each function. $$ y=e^{x+3} $$

Step-by-Step Solution

Answer

The function \( y = e^{x+3} \) is the exponential curve \( y = e^x \) shifted 3 units left.

Step 1: Identify the Basic Function

The given function is \( y = e^{x+3} \). The basic function here is \( y = e^x \), which is the exponential function where the base is Euler's number \( e \approx 2.718 \). This is a well-known curve that passes through the point \( (0,1) \) and has a horizontal asymptote at \( y=0 \).

Step 2: Determine the Transformation

For the function \( y = e^{x+3} \), the term \( x+3 \) inside the exponent indicates a horizontal shift. Specifically, \( x+3 = 0 \) gives \( x = -3 \), meaning that the entire graph of \( y = e^x \) is shifted left by 3 units. This affects the graph by moving all points on \( y = e^x \), including the intercepts and asymptote.

Step 3: Find Key Points

Identify key points on \( y = e^x \) and apply the transformation. For \( y = e^x \), some key points are \( (0, 1) \), \( (1, e) \), and \( (-1, \frac{1}{e}) \). Since the function is shifted left by 3 units, these points on \( y = e^{x+3} \) become \( (-3, 1) \), \( (-2, e) \), and \( (-4, \frac{1}{e}) \).

Step 4: Draw the Asymptote and Plot Points

The horizontal asymptote from \( y = e^x \) at \( y = 0 \) remains the same for \( y = e^{x+3} \). Plot the transformed key points \( (-3, 1) \), \( (-2, e) \), and \( (-4, \frac{1}{e}) \) on a graph. You should notice that the curve passes through these points and asymptotically approaches the x-axis as \( x \) decreases.

Step 5: Sketch the Curve

Draw a smooth exponential curve through the plotted points that approaches the horizontal asymptote \( y=0 \) as \( x \to -\infty \). The graph rises sharply as \( x \to +\infty \). This gives the complete graph of \( y = e^{x+3} \), showing the leftward shift from the base function.

Key Concepts

Graph Transformations

Graph transformations allow us to modify the shapes and positions of standard graphs, such as exponential functions. An exponential function in its base form, like \( y = e^x \), has a distinct, smooth upward curving shape. It passes through specific points and levels off at a horizontal line, known as an asymptote. By applying transformations, we can alter the position and orientation of this graph. For instance, a transformation could include a shift, stretch, or reflection. Transformations are crucial for customizing graphs to adequately represent different algebraic expressions. These changes follow specific rules, making it easier to predict how the graph will appear after applying any transformation.

• Horizontal and vertical shifts
• Stretches and compressions
• Reflections over axes

By mastering graph transformations, you gain powerful tools for tackling various mathematical problems.

Horizontal Shifts

A horizontal shift involves moving a graph left or right across the coordinate plane. This occurs when a constant is added to or subtracted from the independent variable, inside the function's argument. In our example, the function \( y = e^{x+3} \) has a horizontal shift. The term \( x+3 \) suggests that the base function \( y = e^x \) shifts left by 3 units. This is determined by setting \( x+3 = 0 \), solving for \( x \) to get \( x = -3 \). Thus, every point on the graph shifts three units to the left.

• Adding a constant inside: shifts left
• Subtracting a constant inside: shifts right

A horizontal shift doesn't alter the graph's shape. It simply changes its starting position along the x-axis.

Euler's Number

Euler's number, symbolized as \( e \), is approximately equal to 2.718 and is the base of natural logarithms. Named after the mathematician Leonhard Euler, \( e \) is a fundamental constant in mathematics and appears in various formulas across calculus and complex analysis. The natural exponential function, \( y = e^x \), depicts the simplest exponential growth and is significant due to its unique properties. As the base of an exponential function, \( e \) ensures that the curve will always rise steadily without flattening out. The natural exponential graph passes through the point \( (0,1) \) and increases exponentially as \( x \) becomes positive while decreasing toward zero as \( x \) becomes negative.

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. For exponential functions like \( y = e^x \) or \( y = e^{x+3} \), the horizontal asymptote is typically the x-axis, represented as \( y = 0 \). This means that as the x-values decrease dramatically, the y-values get closer but never actually reach zero. The graph gets infinitely close to the asymptote, reflecting the gradual decrease in growth rate of the function. In exponential functions, identifying the asymptote helps in understanding the long-term behavior of the graph. It's especially important in applications such as predicting population growth or radioactive decay, where understanding what values are approached but not exceeded is critical for accurate modeling.