SOLUTION: how do you graph y=(-1/3)(x+1)(x-5)

Question 476107: how do you graph y=(-1/3)(x+1)(x-5)
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
If you look carefully at this problem, you can learn a lot about the general shape of this graph.
.
Notice that by multiplying the two factors together [(x+1)*(x-5)] you get a quadratic equation because you will have an x-squared term, an x-term, and a constant. You probably know that a quadratic equation graphs out as a parabola.
.
Next, you can tell at what points the graph crosses the x-axis. How can you tell? Think about this: what is the value of y for points on the x-axis. y is zero for all points on the x-axis.
.
Next, think about this: When will y = 0 in the equation that you were given? It will be zero if either of the two factors (the two factors are (x + 1) and (x - 5)) are equal to zero. If a factor is equal to zero, then the right side has a multiplier of zero so the entire right side goes to zero and y on the left side will equal the right side value of zero. Look first at the factor (x + 1). When will it equal zero? That will happen if the value of x = -1 because -1 + 1 = 0. So on your graph paper, put a dot on the x-axis at x = -1.
.
Similarly, the other factor (x - 5) will be zero if x = +5. So put a dot on the x-axis at the point x = +5.
.
As you move from left to right on the graph it either drops down to a minimum point and then moving further to the right it rises ... much the same appearance as the path of a skateboarder who comes down a sloped wall to the bottom and then coasts up the opposite wall. Or the path of a snowboarder going from one side of the tube to the other by crossing the bottom.
.
Or you have an alternate possibility. The graph can rise as you go from left to right until it reaches a peak, and then as you continue further to the right it begins to drop. Think of the path of a basketball shot. The ball rises from the shooter's hand, reaches a peak, and then drops back down toward the basket.
.
In any case you can tell which of these two general shapes the quadratic equation gives by looking at the sign of the x-squared term. If the sign is positive, then the graph falls to a minimum point and then rises as you continue to the right. However, if the sign of the x-squared term is negative, then the parabola rises to a peak and then drops back down as you go further to the right.
.
Notice that in this problem the sign on the x-squared term is negative due to the negative sign on the (-1/3) term. It multiplies the x-squared term giving it a minus sign.
.
This being the case, we know that we have a parabola that rises through the left hand point where it crosses the x-axis (at x = -1) rises to a maximum value, and then drops off to cross the x-axis again at x = +5. And it continues going further down as you move further to the right.
.
At what value of x does the peak of the graph occur? It occurs at the point that is midway between the two crossing points on the x-axis. The crossing points are 6 units apart (from -1 to +5 on the x-axis is 6 numbers). The half way point will, therefore, be 3 numbers up from -1 or 3 numbers down from +5. In either case the mid point occurs at x = +2. This means that the peak value of y will occur when x = +2. Substitute +2 for x in the given equation and you get:
.
y = (-1/3)*(x + 1)*(x - 5) which becomes y = (-1/3)*(+2 + 1)*(+2 - 5)
.
This simplifies to:
.
y = (-1/3)*(3)*(-3) = (-1/3)*(-9) = -9/-3 = +3
.
So the peak occurs at the point (+2, +3)
.
You now have enough information to make an approximate graph for the equation:
.
y = (-1/3)*(x + 1)*(x - 5)
.
You can get one more point pretty quickly. Try letting x = 0 and you will find that the value of y (because x equals 0 this will be the y-axis intercept) is +5/3.
.
Now that we've done a quick analysis to give us a rough idea of what the graph looks like, here's a more detailed look:
.

.
Hope this helps you to understand an analytic approach that will aid you in sketching graphs of parabolas.

RELATED QUESTIONS
how do you graph... (answered by Earlsdon)

How do you graph y=... (answered by richwmiller)

How do you graph y= -x + 1... (answered by richwmiller,College Student)

How do you graph... (answered by stanbon)

How do you graph y-x=-1 (answered by josgarithmetic,MathTherapy)

How do you graph y= -3... (answered by josgarithmetic)

How do you graph y=(1/3)^x, and y=3(1/2)^x? (answered by MathLover1)

How do you graph the equation (x+1/2)^2+... (answered by rapaljer)

how do you graph the logarithmic equation y= 1/3^(x-1)... (answered by richard1234)