Graphical Transformations and Finding the Equation of a Curve

In summary, the equation y=(4x+17)/(x+2) undergoes transformation 1 to become y=9/(x+2), transformation 2 to become y=3/(x+2), and transformation 3 to become y=3/(x+4). The original equation of the curve would be y=3/(x+4). In order to sketch a rational function, one should find the asymptotes and intercepts to plot the graph. The vertical asymptote is x=-2 and the horizontal asymptote is y=4. The x-intercept is at (-17/4, 0) and the y-intercept is at (0, 17/2). The resulting graph can be plotted
  • #1
AN630078
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Homework Statement
Good evening, I have a question on graph transformations which I would appreciate some help or guidance with. I have struggled with describing graphical transformations, even though it is a relatively straightforward topic I often confuse myself with the wording of the question.

A curve of y=x^2 undergoes the following transformations in the order given;
1. A translation by -2 units in the direction of the x-axis
2. A stretch by a factor of 3 in the direction of the y-axis
3. A translation of 4 units in the direction of the y-axis

Question a. Give the equation of the resulting curve.
Question b. A second curve undergoes the same three transformations in the same order and the resulting curve is y=4x+17/x+2. Find the equation of the original curve
Question c. Sketch the curve with the equation y=4x+17/x+2
Relevant Equations
y=x^2
a. y=x^2 undergoes transformation 1 to become y=(x+2)^2
y=x^2+2 undergoes transformation 2 to become y=3(x+2)^2
y=3(x+2)^2 undergoes transformation 3 to become y=3(x+2)^2+4

So would the equation of the resulting curve be y=3(x+2)^2+4? I am very uncertain when it comes to performing transformations so I would greatly appreciate any insight or guidance into this problem. I have also graphed and attached an image of y=3(x+2)^2+4 on desmos.

b. If y=4x+17/x+2 to find the original equation of the curve perform the reverse of the transformations in the order 3,2,1.
So would the reversal of these transformations would be a translation of -4 units in the direction of the y-axis, following by a compression of scale factor 3 in the direction of the y-axis and finally a translation of 2 units in the direction of the x-axis (to the left).
y=4x+17/x+2 undergoes this first transformation to become y=4x+17/x+2-4
Transform -4 into a fraction, -4(x+2)/x+2=-4x-8/x+2
y=4x+17/x+2-4 -4x-8/x+2
Since the denominators are equal combine the fractions;
y=4x+17-4x-8/x+2
y=9/x+2

The curve y=9/x+2 undergoes the second transformation, would this be a stretch of scale factor 1/3 in the direction of the y-axis?
So, y=(9/x+2)*1/3
y=9/3(x+2)
y=3/x+2

The curve y=3/x+2 undergoes the third transformation, which I think would be a translation of 2 units in the direction of the x-axis to the left.
Therefore, y=3/(x+2+2), so y=3/x+4

Would the equation of the resulting curve be y=3/x+4

c. In order to sketch a rational function, being y=4x+17/x+2, one should find the asymptotes and the intercepts to plot the graph.
Begin by finding the vertical asymptote, by setting the denominator equal to zero to find any forbidden points.
x+2=0
x=-2
Thus, I cannot have x = -2, and have a vertical asymptote there, which I can sketch on my graph.
Then to find the horizontal asymptote, if the degrees of the numerator and denominator are the same, the horizontal asymptote equals the leading coefficient (the coefficient of the largest exponent) of the numerator divided by the leading coefficient of the denominator.
The horizontal asymptote would be equal to 4/1=4, so I can draw y=4 on my graph also.
Then, find any x- or y-intercepts.
When x=0, y=4(0)+17/0+2, y=17/2
When y=0, 0=4x+17/x+2, 4x+17=0, 4x=-17, x=-17/4

The intercepts are at (0, 17/2) and (-17/4, 0), which I can add to my sketch.
I can then input several points, say from x=-3 to x=3 to form my graph. I have plotted this graph on desmos just to demonstrate how the graph should look. Would this be correct?

I would be very grateful for any help 👍
 

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  • #2
Your answer to the first part looks right. You could check it with a sample x, y.
AN630078 said:
y=4x+17/x+2.
Is that 4x+(17/x)+2, or ((4x+17)/x)+2 or ...?
Similar ambiguities in your woring.
 
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  • #3
haruspex said:
Your answer to the first part looks right. You could check it with a sample x, y.

Is that 4x+(17/x)+2, or ((4x+17)/x)+2 or ...?
Similar ambiguities in your woring.
Thank you for your reply, sorry I should have used brackets.
Question b:
y=(4x+17)/(x+2)
y=(4x+17)/(x+2)undergoes this first transformation to become y=(4x+17)/(x+2-4)
Transform -4 into a fraction, -4(x+2)/(x+2)=(-4x-8)/(x+2)
y=(4x+17)/(x+2)(-4x-8)/(x+2)
Since the denominators are equal combine the fractions;
y=(4x+17-4x-8)/(x+2)
y=9/(x+2)
The curvey=9/(x+2) undergoes the second transformation, y=9/(x+2)*(1/3)
y=9/(3(x+2))
y=3/(x+2)
The curve y=3/(x+2) undergoes the third transformation,therefore, y=3/(x+2+2), so y=3/(x+4)

The equation of the original curve would be y=3/(x+4)

Also, would my method and resultant curve for y=(4x+17)/(x+2) be correct in question c?
 
  • #4
In solving a), your first step replaced x with x+2. What should your final step be in b)?

Your graph for c looks right.
 

1. What are graphical transformations?

Graphical transformations refer to the changes made to a graph of a function, such as shifting, stretching, or reflecting the graph. These transformations are represented by changes in the equation of the curve.

2. How do I determine the equation of a curve from a graph?

To find the equation of a curve from a graph, you need to identify key points on the graph, such as the vertex, intercepts, and any other known points. Then, use these points to create a system of equations and solve for the coefficients of the equation.

3. What is the difference between a horizontal and vertical shift?

A horizontal shift involves moving the graph of a function left or right along the x-axis, while a vertical shift involves moving the graph up or down along the y-axis. These shifts can be represented by adding or subtracting a constant value to the original equation.

4. How do I determine the type of transformation from a given equation?

The type of transformation can be determined by looking at the coefficients in the equation. A change in the coefficient of x will result in a horizontal transformation, while a change in the coefficient of y will result in a vertical transformation. Additionally, a negative coefficient will result in a reflection of the graph.

5. Can graphical transformations affect the shape of a curve?

Yes, graphical transformations can affect the shape of a curve. For example, a horizontal stretch or compression will result in a wider or narrower curve, while a vertical stretch or compression will result in a taller or shorter curve. Reflections can also change the orientation of the curve.

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