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Sagot :
To draw the graph of the function [tex]\( f(x) = -2^{x+4} \)[/tex], we need to follow a series of steps to understand the function's behavior and accurately depict it. Here's a detailed, step-by-step solution:
### Step 1: Understanding the Function
The given function is [tex]\( f(x) = -2^{x+4} \)[/tex]. Here:
- The base of the exponent is 2.
- The exponent is [tex]\( x + 4 \)[/tex].
- The negative sign in the front indicates that the function is reflected over the x-axis.
### Step 2: Exponential Behavior
Exponential functions of the form [tex]\( 2^{x+4} \)[/tex] grow rapidly as [tex]\( x \)[/tex] increases and decay towards zero as [tex]\( x \)[/tex] decreases. However, the negative sign will invert this behavior.
### Step 3: Shifts and Transformations
- The term [tex]\( x + 4 \)[/tex] indicates a horizontal shift. The function [tex]\( 2^x \)[/tex] is shifted to the left by 4 units.
- The negative sign indicates that each value of [tex]\( 2^{x+4} \)[/tex] is multiplied by -1.
### Step 4: Evaluating at Key Points
Let's evaluate the function at several key points to get a sense of the graph:
- [tex]\( f(-5) = -2^{-5+4} = -2^{-1} = -\frac{1}{2} = -0.5 \)[/tex]
- [tex]\( f(-4) = -2^{-4+4} = -2^0 = -1 \)[/tex]
- [tex]\( f(-3) = -2^{-3+4} = -2^1 = -2 \)[/tex]
- [tex]\( f(-2) = -2^{-2+4} = -2^2 = -4 \)[/tex]
- [tex]\( f(-1) = -2^{-1+4} = -2^3 = -8 \)[/tex]
- [tex]\( f(0) = -2^{0+4} = -2^4 = -16 \)[/tex]
- [tex]\( f(1) = -2^{1+4} = -2^5 = -32 \)[/tex]
As we can observe, the function quickly decreases in value as [tex]\( x \)[/tex] increases. When [tex]\( x \)[/tex] is sufficiently negative (large negative values), the function becomes very small (close to zero).
### Step 5: Plotting Points
Now, we plot these points on a coordinate system:
- [tex]\( (-5, -0.5) \)[/tex]
- [tex]\( (-4, -1) \)[/tex]
- [tex]\( (-3, -2) \)[/tex]
- [tex]\( (-2, -4) \)[/tex]
- [tex]\( (-1, -8) \)[/tex]
- [tex]\( (0, -16) \)[/tex]
- [tex]\( (1, -32) \)[/tex]
### Step 6: Sketching the Graph
- The curve approaches zero as [tex]\( x \to -\infty \)[/tex] but never actually reaches zero.
- The curve decreases rapidly as [tex]\( x \)[/tex] increases.
### Graph Summary
The graph of [tex]\( f(x) = -2^{x+4} \)[/tex] will:
- Be a downward curve starting from values close to zero for large negative [tex]\( x \)[/tex],
- Pass through calculated points we evaluated,
- Decrease rapidly as [tex]\( x \)[/tex] increases.
### Final Graph
To show the complete graph:
1. Draw the x and y axes.
2. Mark the key points evaluated.
3. Sketch a smooth curve through the points, approaching zero on the left and rapidly declining on the right.
The completed plot will show the function’s exponential decay inverted due to the negative sign, shifted horizontally due to the [tex]\( x + 4 \)[/tex] term.
### Step 1: Understanding the Function
The given function is [tex]\( f(x) = -2^{x+4} \)[/tex]. Here:
- The base of the exponent is 2.
- The exponent is [tex]\( x + 4 \)[/tex].
- The negative sign in the front indicates that the function is reflected over the x-axis.
### Step 2: Exponential Behavior
Exponential functions of the form [tex]\( 2^{x+4} \)[/tex] grow rapidly as [tex]\( x \)[/tex] increases and decay towards zero as [tex]\( x \)[/tex] decreases. However, the negative sign will invert this behavior.
### Step 3: Shifts and Transformations
- The term [tex]\( x + 4 \)[/tex] indicates a horizontal shift. The function [tex]\( 2^x \)[/tex] is shifted to the left by 4 units.
- The negative sign indicates that each value of [tex]\( 2^{x+4} \)[/tex] is multiplied by -1.
### Step 4: Evaluating at Key Points
Let's evaluate the function at several key points to get a sense of the graph:
- [tex]\( f(-5) = -2^{-5+4} = -2^{-1} = -\frac{1}{2} = -0.5 \)[/tex]
- [tex]\( f(-4) = -2^{-4+4} = -2^0 = -1 \)[/tex]
- [tex]\( f(-3) = -2^{-3+4} = -2^1 = -2 \)[/tex]
- [tex]\( f(-2) = -2^{-2+4} = -2^2 = -4 \)[/tex]
- [tex]\( f(-1) = -2^{-1+4} = -2^3 = -8 \)[/tex]
- [tex]\( f(0) = -2^{0+4} = -2^4 = -16 \)[/tex]
- [tex]\( f(1) = -2^{1+4} = -2^5 = -32 \)[/tex]
As we can observe, the function quickly decreases in value as [tex]\( x \)[/tex] increases. When [tex]\( x \)[/tex] is sufficiently negative (large negative values), the function becomes very small (close to zero).
### Step 5: Plotting Points
Now, we plot these points on a coordinate system:
- [tex]\( (-5, -0.5) \)[/tex]
- [tex]\( (-4, -1) \)[/tex]
- [tex]\( (-3, -2) \)[/tex]
- [tex]\( (-2, -4) \)[/tex]
- [tex]\( (-1, -8) \)[/tex]
- [tex]\( (0, -16) \)[/tex]
- [tex]\( (1, -32) \)[/tex]
### Step 6: Sketching the Graph
- The curve approaches zero as [tex]\( x \to -\infty \)[/tex] but never actually reaches zero.
- The curve decreases rapidly as [tex]\( x \)[/tex] increases.
### Graph Summary
The graph of [tex]\( f(x) = -2^{x+4} \)[/tex] will:
- Be a downward curve starting from values close to zero for large negative [tex]\( x \)[/tex],
- Pass through calculated points we evaluated,
- Decrease rapidly as [tex]\( x \)[/tex] increases.
### Final Graph
To show the complete graph:
1. Draw the x and y axes.
2. Mark the key points evaluated.
3. Sketch a smooth curve through the points, approaching zero on the left and rapidly declining on the right.
The completed plot will show the function’s exponential decay inverted due to the negative sign, shifted horizontally due to the [tex]\( x + 4 \)[/tex] term.
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