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How to Sketch the Transformed Exponential Function $y=2^{x+3}$
Mathematics
Grade 11 (Senior High School)
Question Content
Sketch the graph of the function $y = 2^{x+3}$
Correct Answer
A horizontal shift left by 3 units of the parent exponential function $y=2^x$, with y-intercept at $(0,8)$, horizontal asymptote at $y=0$, passing through point $(-3,1)$
Detailed Solution Steps
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Step 1: Identify the parent function. The given function is a transformation of the basic exponential parent function $y = 2^x$, which has a horizontal asymptote at $y=0$, passes through $(0,1)$ and $(1,2)$, and increases as $x$ increases.
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Step 2: Analyze the transformation. The exponent is $x+3$, which follows the form $y = b^{x+h}$. For exponential functions, a value $h$ added directly to $x$ inside the exponent causes a horizontal shift: if $h>0$, the graph shifts left by $h$ units. Here $h=3$, so we shift the graph of $y=2^x$ left 3 units.
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Step 3: Find key points for the transformed function. For the parent function $y=2^x$, the point $(0,1)$ shifts left 3 units to $(-3,1)$. The y-intercept is found by setting $x=0$: $y=2^{0+3}=2^3=8$, so the y-intercept is $(0,8)$. The horizontal asymptote remains $y=0$ because horizontal shifts do not affect the horizontal asymptote of exponential functions.
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Step 4: Sketch the graph. First draw the horizontal asymptote $y=0$. Plot the key points $(-3,1)$ and $(0,8)$, then draw a smooth, increasing curve that approaches $y=0$ as $x$ approaches negative infinity and rises rapidly as $x$ approaches positive infinity, matching the shape of the parent function but shifted left 3 units.
Knowledge Points Involved
1
Basic Exponential Functions
Exponential functions have the form $y = b^x$ where $b>0$, $b\\neq1$. When $b>1$ (like $b=2$ here), the function is an increasing exponential growth function, with a horizontal asymptote at $y=0$, passing through the point $(0,1)$. These functions model exponential growth scenarios like compound interest or population growth.
2
Horizontal Transformations of Functions
For any function $f(x)$, the transformation $f(x+h)$ results in a horizontal shift of the original graph. If $h>0$, the graph shifts left by $h$ units; if $h<0$, the graph shifts right by $|h|$ units. This transformation affects the input values (x-values) of the function, rather than the output (y-values).
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Finding Intercepts of Functions
The y-intercept of a function is found by setting $x=0$ and solving for $y$, as it is the point where the graph crosses the y-axis. For exponential functions of the form $y=b^{x+h}$, the y-intercept is $y=b^h$. Horizontal asymptotes of exponential functions are not affected by horizontal shifts, remaining at $y=0$ for functions of the form $y=b^{x+h}+k$ when $k=0$.
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