How do you graph the function #y=5x-x^2#?

Question

Paisley Johnson · Accepted Answer

graph{5x-x^2 [-14.24, 14.23, -7.12, 7.12]} First you realise that #5x-x^2=x(5-x)# which will give the
#y=0# solutions #x=0# and #x=5# for #(0,0)# and #(5,0)# Then, put the halfway-#x# (#2.5#) in the equation to get the top of the graph #y=5*2.5-2.5^2=6.25# for #(2.5,6.25)#  You now have the three most important points of your graph. You can put more #x#-values in, of course, so you get  more points, but these three are essential.

Adam Adkins · Answer

undefined To graph the function $ y = 5x - x^2 $, follow these steps:

1. Identify the key components of the function: the coefficient of the linear term (5), and the coefficient of the quadratic term (-1).

2. Determine the vertex of the parabola using the formula for the vertex of a quadratic function: $ \left( \frac{-b}{2a}, f\left(\frac{-b}{2a}\right) \right) $, where $ a = -1 $ and $ b = 5 $.

3. Calculate the $ y $-intercept by setting $ x = 0 $ and solving for $ y $.

4. Find the $ x $-intercepts by setting $ y = 0 $ and solving the resulting quadratic equation.

5. Plot the vertex, $ y $-intercept, and $ x $-intercepts on the coordinate plane.

6. Since the coefficient of $ x^2 $ is negative, the parabola opens downward.

7. Draw the parabola passing through the plotted points, with its vertex as the highest or lowest point depending on the sign of the coefficient of $ x^2 $.

8. Label the axes and any important points on the graph for clarity.