Preliminaries Page 3c

Page 3c

Other Information About the Parabola

EXAMPLE (Continued from Page 3a)

Consider the parabola defined by the equation y = (1/4)(x - 6)² + 3.
After graphing this parabola answer the following questions:

What is the axis of symmetry?
What is the domain?
What is the range?
What are the x-intercepts, if any exist?
What are the y-intercepts, if any exist?
FIGURE 1

You have already graphed this parabola and arrived at the following:

Axis of symmetry: Since this parabola has its vertex at the point (6, 3), and it opens up, you can conlude that the parabola is symmetric about the vertical line running through the vertex.
That is, this parabola is symmetric about the line x = 6.

Domain: The domain of a function is the set of values that the independent variable (the x variable) can take on. In this case, the function is defined for every value of x.
Therefore the domain is the entire number line: {x: -¥ < x < ¥ }

Range: The range of a function is the set of values that the function takes on. That is, all values of y which can be obtained by entering the values of x from the domain. Since this parabola increases forever from the vertex, the range is the set of all y values greater than or equal to 3: {y: y ³ 3}

x-intercepts: By looking at the graph, you can easily see that this parabola has no x-intercepts.
This is the same as saying that there are no roots or zeros of the equation y = (1/4)(x - 6)² + 3.
You can verify this observation by trying to solve the equation: y = (1/4)(x - 6)² + 3 = 0
==>(1/4)(x - 6)² = -3
==>(x - 6)² = -12
Since there is no real number with a negative square, you can conclude that this equation has no solutions.
Therefore, this parabola has no x-intercepts.

y-intercepts: You can clearly see that this graph has one y-intercept. By zooming in you can determine that this interception point is at y = 12. Let's verify this algebraically.
To find the y-intercept, plug x = 0 into the equation. y = (1/4)(0 - 6)² + 4
==> y = (1/4)36 + 3
==> y = 9 + 3
==> y = 12 Therefore the y-intercept is at y = 12

EXAMPLE (Continued from Page 3a)
Consider the parabola defined by the equation y = (1/4)(x - 6)² + 3. After graphing this parabola answer the following questions: What is the axis of symmetry? What is the domain? What is the range? What are the x-intercepts, if any exist? What are the y-intercepts, if any exist?	FIGURE 1
You have already graphed this parabola and arrived at the following:	FIGURE 1

Axis of symmetry:	Since this parabola has its vertex at the point (6, 3), and it opens up, you can conlude that the parabola is symmetric about the vertical line running through the vertex. That is, this parabola is symmetric about the line x = 6.
Domain:	The domain of a function is the set of values that the independent variable (the x variable) can take on. In this case, the function is defined for every value of x. Therefore the domain is the entire number line: {x: -¥ < x < ¥ }
Range:	The range of a function is the set of values that the function takes on. That is, all values of y which can be obtained by entering the values of x from the domain. Since this parabola increases forever from the vertex, the range is the set of all y values greater than or equal to 3: {y: y ³ 3}
x-intercepts:	By looking at the graph, you can easily see that this parabola has no x-intercepts. This is the same as saying that there are no roots or zeros of the equation y = (1/4)(x - 6)² + 3. You can verify this observation by trying to solve the equation: y = (1/4)(x - 6)² + 3 = 0 ==>(1/4)(x - 6)² = -3 ==>(x - 6)² = -12 Since there is no real number with a negative square, you can conclude that this equation has no solutions. Therefore, this parabola has no x-intercepts.
y-intercepts:	You can clearly see that this graph has one y-intercept. By zooming in you can determine that this interception point is at y = 12. Let's verify this algebraically. To find the y-intercept, plug x = 0 into the equation. y = (1/4)(0 - 6)² + 4 ==> y = (1/4)36 + 3 ==> y = 9 + 3 ==> y = 12 Therefore the y-intercept is at y = 12