To graph the function [tex]\( y = 3 \cdot 7^{-x} + 2 \)[/tex] using transformations, follow these step-by-step instructions:
1. Start with the basic exponential function [tex]\( y = 7^x \)[/tex]:
- The graph of [tex]\( y = 7^x \)[/tex] is an exponential growth curve that passes through the point [tex]\( (0, 1) \)[/tex] and increases rapidly as [tex]\( x \)[/tex] increases. This is our base function.
2. Reflect this graph over the y-axis to get [tex]\( y = 7^{-x} \)[/tex]:
- To reflect the graph over the y-axis, we replace [tex]\( x \)[/tex] with [tex]\( -x \)[/tex]. The graph of [tex]\( y = 7^{-x} \)[/tex] is a decreasing exponential function that passes through the point [tex]\( (0, 1) \)[/tex] and decreases towards zero as [tex]\( x \)[/tex] increases. This reflection changes the direction of the growth, turning it into a decay.
3. Stretch the graph vertically by a factor of 3 to get [tex]\( y = 3 \cdot 7^{-x} \)[/tex]:
- To stretch the graph vertically, multiply the function by 3. This alters the graph so that for any given [tex]\( x \)[/tex], the value of [tex]\( y \)[/tex] is tripled. This means the graph now passes through the point [tex]\( (0, 3) \)[/tex] and retains its general decreasing shape but is stretched taller.
4. Translate the graph upwards by 2 units to get [tex]\( y = 3 \cdot 7^{-x} + 2 \)[/tex]:
- To translate the graph upwards, add 2 to the entire function. This means that every point on the graph is moved up by 2 units. The new graph now passes through the point [tex]\( (0, 5) \)[/tex]. The horizontal asymptote of the function also shifts from [tex]\( y = 0 \)[/tex] to [tex]\( y = 2 \)[/tex].
To summarize, the resulting graph of [tex]\( y = 3 \cdot 7^{-x} + 2 \)[/tex] can be visualized through the following transformations applied to the basic exponential function [tex]\( y = 7^x \)[/tex]:
- Reflect over the y-axis.
- Stretch vertically by a factor of 3.
- Translate upwards by 2 units.
These transformations give us the final graph which is a vertically stretched and upwards shifted version of the reflected exponential decay.
