Question Graph the function $$ f(x)=7^{-x} $$ by moving key points. You can only graph integer points, for example you cannot plot the point $$ \left(-1, \frac{1}{4}\right) $$.

Question

UpStudy AI · Accepted Answer

1. Note that the function can be written as  
   $$
   f(x)=7^{-x} = (1/7)^x.
   $$

2. The graph of $$7^x$$ has easy-to-plot integer points:  
   $$
   (0,7^0)=(0,1),\quad (1,7^1)=(1,7),\quad (2,7^2)=(2,49),\quad (3,7^3)=(3,343),\quad \ldots
   $$

3. Since  
   $$
   f(x)=7^{-x}=7^{(-x)},
   $$  
   its graph is the reflection of the graph of $$7^x$$ in the $$y$$-axis. To find key points for $$f(x)$$, replace $$x$$ with $$-x$$ in the points above.

4. Reflecting the points over the $$y$$-axis gives:  
   - The point $$(0,1)$$ remains the same.  
   - The point $$(1,7)$$ becomes $$(-1,7)$$.  
   - The point $$(2,49)$$ becomes $$(-2,49)$$.  
   - The point $$(3,343)$$ becomes $$(-3,343)$$.

5. These are all integer points! So you can plot the points:  
   $$
   (-3,343),\quad (-2,49),\quad (-1,7),\quad (0,1).
   $$

6. Finally, note that as $$x$$ increases from 0 (i.e. for $$x>0$$),  
   $$
   f(x)=(1/7)^x,
   $$  
   which approaches 0. Therefore, the horizontal line  
   $$
   y=0
   $$  
   is the horizontal asymptote of the function.

The graph is obtained by plotting these integer key points and drawing a smooth curve through them, reflecting exponential growth for negative $$x$$ (with rapidly increasing values) and exponential decay for positive $$x$$ (approaching 0).
To graph $$ f(x) = 7^{-x} $$ using integer points: 1. Start with the points from $$ 7^x $$: - $$ (0,1) $$ - $$ (1,7) $$ - $$ (2,49) $$ - $$ (3,343) $$ 2. Reflect these points over the y-axis to get: - $$ (-3,343) $$ - $$ (-2,49) $$ - $$ (-1,7) $$ - $$ (0,1) $$ 3. Plot these integer points and draw a smooth curve through them. 4. Note that as $$ x $$ increases from 0, $$ f(x) $$ approaches 0, making the horizontal line $$ y=0 $$ an asymptote. This gives the graph of $$ f(x) = 7^{-x} $$ using only integer points.

Question
Graph the function by moving key points.
You can only graph integer points, for example you cannot plot the point .

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Pre Calculus Questions

Question Graph the function by moving key points. You can only graph integer points, for example you cannot plot the point .