How do you graph the function and its inverse of #f(x)=-(x-3)^2+1#?

Question

Lily Anderson · Accepted Answer

Maximum turning point at (3,1) and y-intercept at (0,-8)

Let #y=-(x-3)^2+1# The curve has a minimum point at #(3, 1)# since this is the completed square form. #y=-(x^2-6x+9)+1# #y=-x^2+6x-8# The curve is a 'n' shaped quadratic since the coefficient of #x^2# is negative, so the turning point is a maximum. let #x=0, y=-8# From this information, we can put a turning point at #(3,1)# and a y-intercept at #y=-8#. graph{-(x-3)^2+1 [-11.25, 11.25, -5.63, 5.62]} now for #y=f^-1(x)# Looking at f(x), for our input x, we go: #x-> -3 -> Ans^2 -> xx-1 -> +1 -> f(x)# So #f(x) -> -1 -> -:-1 -> sqrtAns -> +3 -> f^-1(x)# So #f^-1(x)=sqrt(-(x-1))+3# Let #y=sqrt(-(1-x))+3 # Immediately we get that part of the graph will be imaginary. This part of the graph is where the bit under the square root sign #(-(x-1))<=0# So let #-(x-1)>=0# #1-x>=0# #1>=x# #x<=1# So any x values above 1 aren't plotted. Let #y=0# #0=sqrt(1-x)+3# #-3=sqrt(1-x)# #9=1-x# #x=-8# But subbing this into #f^-1(x)# gives us #sqrt(1--8)+3=sqrt9+3=6!=0# So we have no x-intercepts. The graph starts when x=1. So let #x=1# #y=sqrt(1-1)+3# #y=3# So we start the graph at (1,3). To find the y-intercept, let #x=0# #y=sqrt1+3=4# Now we draw our graph, starting at (1,3) and passing through (4,0) graph{sqrt(-(x-1))+3 [-13.875, 8.625, -0.995, 10.255]}

Xavier Alonso · Answer

undefined To graph the function $ f(x) = -(x - 3)^2 + 1 $ and its inverse, follow these steps:

1. Plot points for the function:
   - Choose several x-values.
   - Calculate the corresponding y-values using the function equation.
   - Plot these points on the graph.

2. Draw the graph of the function by connecting the plotted points smoothly.

3. To find the inverse of the function:
   - Replace $ f(x) $ with $ y $.
   - Interchange $ x $ and $ y $.
   - Solve for $ y $ to express $ y $ in terms of $ x $.

4. Plot points for the inverse function using the same method as for the original function.

5. Draw the graph of the inverse function by connecting the plotted points smoothly.

Ensure that both graphs are accurately labeled and clearly distinguishable from each other on the coordinate plane.

Keep in mind that the inverse of a function reflects across the line $ y = x $.