Plotting Quadratic Equations

Figure 1

Shape of the Quadratic Equation and Turning Points

When we draw quadratic equations in the form y = ax² + bx + c, the shape of the curve is either a ∪ or ∩ shape. We can tell what shape the quadratic equation form without plotting points on a 2-dimensional graph by looking at the a (the coefficient of x²). If a is positive, then shape of the quadratic equation is a ∪ shape, if a is negative then the shape of the quadratic equation is a ∩ shape.
You can see that both ∪ and ∩ shapes have a turning point. The turning point is the point on the curve at which the curve's direction changes. Look at figure 1. To understand what the turning point is and how to find it, you need to understand how to complete the square. Please check out the completing square part 1 page and completing the square part 2 page first before you understand this page.
In figure 1, the quadratic equation y = x² + 2x + 1 has been drawn for when x equals -3 to 1. The coefficient of x² is 1 (which is a positive number), so the shape of the curve would be a ∪ shape; you can see this in figure 1.
So the turning point of the curve is easy to identify on a 2-dimensional graph because you can see it. We can see that the curve's turning point is at the point (-1,0). You can see that the turning point of the curve in figure 1 is the lowest point on the curve. What does this mean?
When x (the input) = -1, y (the output) is equal to 0. So the smallest value that the output could be is 0, as (-1,0) is the lowest point on the curve. THUS, THE TURNING POINT, IN THIS CASE, CAN BE CALLED THE MINIMUM POINT AS WELL.

Figure 2

In figure 2, the quadratic equation y = -2x² + 4x - 3 has been drawn for when x equals 0 to 2. The coefficient of x² is -2 (which is a negative number), so the shape of the curve would be a ∩ shape; you can see this in figure 2.
So the turning point of the curve is easy to identify on a 2-dimensional graph because you can see it. We can see that the curve's turning point is at the point (1,-1). You can see that the turning point of the curve in figure 2 is the highest point on the curve. What does this mean?
When x (the input) = 1, y (the output) is equal to -1. So the highest value that the output could be is -1, as (1,-1) is the highest point on the curve. THUS, THE TURNING POINT, IN THIS CASE, CAN BE CALLED THE MAXIMUM POINT AS WELL.
Identifying turning points when the they are already drawn on a graph is not hard because you can see them. What do we if we need to draw a quadratic equation for when x (the input) takes on a certain range of values. We would substitute x (the input) with each value to find the value of the output, and thus we can plot the coordinates. However, we would still need to find the turning point. This is discussed below.

Finding the Turning Point

Lets imagine that we did not have figure 1 and figure 2.
Lets first find the turning point on the curve of y = x² + 2x + 1.
We have to express x² + 2x + 1 in the form m(x + y)² - e. To do this you need to know how to complete the square.
x² + 2x + 1 in the form (x + y)² - e is (x + 1)² + 0, so m = 1, y = 1 and e = 0
So we can write y = x² + 2x + 1 = (x + 1)² .
We need to look closer at the equation y = (x + 1)² .
x + 1 is inside the brackets, if the value of (x + 1) is positive and we square that number, a positive number squared is a positive number.
If the value of (x + 1) is negative and we square that number, a negative number squared is a positive number.
So the value of (x + 1)² would always be bigger than 0 for any values of x, except for when x = -1. When x = -1, then (-1 + 1)²= 0. This means that the lowest value of y (the output) is 0 and this happens when x = -1. As this the lowest value that y (the output) can be, (-1,0) is the lowest point of the curve.
Thus (-1,0) is the turning point point on the curve of y = x² + 2x + 1.
Lets now find the turning point of the curve y = -2x² + 4x - 3.
We have to express -2x² + 4x - 3 in the form m(x + y)² - e.
-2x² + 4x - 3 in the form m(x + y)² - e is -2(x - 1)² - 1, so m = -2, y = -1 and e = 1
So we can write y = -2x² + 4x - 3 = -2(x - 1)² - 1.
We need to look closer at the equation y = -2(x - 1)² - 1
Lets focus on the -2(x - 1)² part of the equation.
If the value of (x -1) is positive and we square that number, a positive number squared is a positive number.
If the value of (x -1) is negative and we square that number, a negative number squared is a positive number.
This positive number we attain is multiplied by -2 (which is a negative number); this means that the value of -2(x - 1)² would always be less than 0, unless x = 1.
So the value of -2(x - 1)² - 1 is biggest when the -2(x - 1)² part is equal to 0; this happens when x = 1. When x (the input) = 1, y (the output) = -1, as shown below.
y = -2(1 - 1)² - 1 = -2(0) - 1 = 0 - 1 = -1
When x does not equal 1, the value of -2(x - 1)² would be a negative number, and thus the number would be less than 0. 1 is then subtracted from this number.
when x = 1, y = 0 - 1 = -1when x ≠ 1, y = A number less than 0 - 1
You can see that when x does not equal 1, the y value becomes less than -1. This means that the highest value of y (the output) is -1.
This means that the turning point on the curve is (1,-1). In this case (1,-1) is called the maximum point as well.
So now you should understand the importance of knowing how to complete the square and why it is important.

Question

Find the values of the function f(x) = x² + 3x - 2, for when x is substituted with values -4 to 1. Plot the function, for the values of x from -3 to 3 including the turning point.
So we need to substitute x with the following values, -3, -2, -1, 0, 1, 2 and 3.when x = -4, f(-4) = (-4)² + 3(-4) - 2 = 16 - 12 - 2 = 2when x = -3, f(-3) = (-3)² + 3(-3) - 2 = 9 - 9 - 2 = -2when x = -2, f(-2) = (-2)² + 3(-2) - 2 = 4 - 6 - 2 = -4when x = -1, f(-1) = (-1)² + 3(-1) - 2 = 1 - 3 - 2 = -4when x = 0, f(0) = (0)² + 3(0) - 2 = 0 + 0 - 2 = -2when x = 1, f(1) = (1)² + 3(1) - 2 = 1 + 3 - 2 = 2
Using coordinates we can represent our inputs and outputs on a 2 dimensional graph; we can represent the function f(x) for when x takes on values from -3 to 1.
We can write the inputs and outputs as coordinates. Once we attain coordinates, we are able to plot them. The inputs are the x values and the outputs are the values of f(x). You can see above when we input the values of x from -3 to 1, we get outputs, so lets now write the inputs and outputs in form of (x,f(x)) which is in coordinate form. x is the input and f(x) is the output, thus horizontal axis has the x values on it and the vertical axis has the output values on it, in this case the value of f(x).
when x = -4, f(-4) = 2 so in coordinate form we can write (-4,2)when x = -3, f(-3) = -2 so in coordinate form we can write (-3,-2)when x = -2, f(-2) = -4, so in coordinate form we can write (-2,-4)when x = -1, f(-1) = -4, so in coordinate form we can write (-1,-4)when x = 0, f(0) = -2, so in coordinate form we can write (0,-2)when x = 1, f(1) = 2, so in coordinate form we can write (1,2)
We get
(-4, 2), (-3,-2), (-2,-4), (-1,-4), (0,-2), (1,2)
As shown in figure 3, we can plot these points.
We also have to plot the turning point. To find the turning point, we can use what we have learned from the completing the square pages.
Lets express x² + 3x - 2 in the form (x + y)² - e.
By completing the square we get, (x + 3/2)² - 4.25
-3/2 = -1.5 (in decimal form)
So our turning point is (-1.5, -4.25)

Thus we have plotted the function f(x) = x² + 3x - 2 , when x takes on values from -3 to 3, as shown in figure 4.

Figure 3

Figure 4

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