Draw the graph of y= 3x( 4-x) for the values of x ranging from -2 to +6. On the same axes draw the line y = 5(x-2). Use a scale of 1cm to 1 unit on the x axis and 1 cm to 5 units on the y axis. From the graph,
A. Find the roots of the equation 3x(4-x)=0,
B. Write down the maximum value of y=3x(4-x)
C. Deduce the roots of the equation 3x(4-x)=5(x-2)
To draw the graph of y = 3x(4-x), we can first find the x and y values for several points by substituting different values of x into the equation.
When x = -2:
y = 3(-2)(4-(-2)) = -18
When x = -1:
y = 3(-1)(4-(-1)) = -15
When x = 0:
y = 3(0)(4-0) = 0
When x = 1:
y = 3(1)(4-1) = 9
When x = 2:
y = 3(2)(4-2) = 12
When x = 3:
y = 3(3)(4-3) = 9
When x = 4:
y = 3(4)(4-4) = 0
When x = 5:
y = 3(5)(4-5) = -15
When x = 6:
y = 3(6)(4-6) = -18
Using these points, we can plot the graph as shown below:
```
| x
20 | *
| *
15 | *
| *
10 | *
|*
5 |_____ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
-2 0 2 4 6
```
Next, let's draw the line y = 5(x-2) on the same axes.
To find the x and y values for the line, we can substitute different values of x into the equation.
When x = -2:
y = 5(-2-2) = -20
When x = -1:
y = 5(-1-2) = -15
When x = 0:
y = 5(0-2) = -10
When x = 1:
y = 5(1-2) = -5
When x = 2:
y = 5(2-2) = 0
When x = 3:
y = 5(3-2) = 5
When x = 4:
y = 5(4-2) = 10
When x = 5:
y = 5(5-2) = 15
When x = 6:
y = 5(6-2) = 20
Using these points, we can plot the line as shown below:
```
| x
20 | *
| *
15 | *
| *
10 | *
|*
5 |__________
-2 0 2 4 6
```
A. To find the roots of the equation 3x(4-x) = 0, we need to find the values of x where y = 0. From the graph, we can see that the two roots are x = 0 and x = 4.
B. To find the maximum value of y = 3x(4-x), we can look at the highest point on the graph. From the graph, we can see that the maximum value occurs at x = 2, where y = 12.
C. To deduce the roots of the equation 3x(4-x) = 5(x-2), we need to find the values of x where the graph of y = 3x(4-x) intersects the graph of y = 5(x-2). From the graph, we can see that the two intersecting points are x = 0 and x = 4, which coincide with the roots obtained in part A.