17 ## TRANSLATIONSVertical stretches and shrinks A TRANSLATION OF A GRAPH is its rigid movement, vertically or horizontally. On the left is the graph of the absolute value function. On the right is its translation to a "new origin" at (3, 4). The equation of the absolute value function is
The equation of its translation to (3, 4) is
For, when Thus the point (3, 4) is that point on the translated graph that was originally at (0, 0). In general,
When We will prove that below. Example 1. Write the equation of this graph: The graph of the absolute value has been translated 3 units up, but 5 units to the
Problem 1. Write the equation of this graph: To see the answer, pass your mouse over the colored area.
Not only has the graph of the absolute value been translated, it has first been reflected about the A translation is a rigid movement of the graph. The
and then wrote the relection about the
that would be wrong. You could see that because when Problem 2. Sketch the graph of
Problem 3. Sketch the graph of
Problem 4. Sketch the graph of
This is equivalent to The graph is reflected about the
Problem 5. Sketch the graph of This is the square root function translated 1 unit to the right.
Problem 6. Sketch the graph of This is the reflected square root function, translated 3 units to the left.
Problem 7. Sketch the graph of This is equivalent to
Example 2. The vertex of a parabola. Write the equation of the parabola (with leading coefficient 1) whose vertex is at the point ( Problem 8. Write the equation of the parabola whose vertex is at
Example 3. What are the coördinates of the vertex of this parabola?
The vertex will then be at ( Now,
Therefore, Example 4. What are the coördinates of the vertex of this parabola?
If we simply transpose 5 --
-- we see that Example 5. Completing the square. What are the coördinates of the vertex of this parabola?
-- we will transpose the constant term, and complete the square on the right.
The vertex is at (−3, −11). Problem 9. What are the coördinates of the vertex of this parabola?
The right-hand side is the perfect square of (
The vertex therefore is at (5, 0). Problem 10. What are the coördinates of the vertex of this parabola?
The equation implies
The vertex is at (0 −1). Problem 11. What are the coördinates of the vertex of this parabola?
Transpose the constant term, and complete the square on the right:
The vertex is at (4, −15). The equation of a circle What characterizes every point ( Every point (
This is the equation of a circle of radius Specifically, this --
-- is the equation of a circle of radius 5 centered at the origin. Every pair of values (
Question. What is the equation of a circle with center at ( The circle has been translated from (0, 0) to ( Problem 12. Write the equation of the circle of radius 3, and center at the following point.
Example 6. Show that the following is the equation of a circle. Name the radius and the coördinates of the center.
( Therefore, we will complete the square in both To complete the square in To complete the square in
This is the equation of a circle of radius 4, whose center is at (2, 1). We can say then that when a quadratic in
-- then that is the equation of a circle. The coefficients of Problem 13. Show that the following is the equation of a circle. Name the radius and the coördinates of the center.
Transpose the constant term, and complete the square in both
This is the equation of a circle of radius 6, with center at (−3, −5). Here is the proof of the main theorem. Theorem.
For in a translation, every point on the graph moves in the same manner. Let (
And let us translate the graph
and
If Now, what will be the equation of the translated graph, such that when the value of We say that the following is the equation:
For, when
And (
Which is what we wanted to prove. Vertical stretches and shrinks If we multiply a function If we multiply But if we multiply Next Topic: Rational functions Please make a donation to keep TheMathPage online. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com |