15 decreased by twice a number
2 answers:
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The graph of y = 2(x - 3)² + 2 can be seen in the attached picture. This problem can be solved through the concept of parabola and transformation.
<h3>Further explanation</h3><u>The Problem:</u>
Which is the graph of y = 2(x - 3)² + 2?
<u>Question-1:</u>
How to make a graph y = 2 (x - 3) ² + 2 through the concept of a parabola.
<u>The Process:</u>
The equation of a parabola is given by .
Keep in mind the following points:
- vertex point at (h, k)
- axis of symmetry at x = h
- a > 0 the parabola opens upward
- a < 0 the parabola opens downward
- the y-intercept is
at x = 0.
From our case it can be concluded as follows:
- the graph of y = 2(x - 3)² + 2 opens upward
- vertex point at (3, 2)
- axis of symmetry at x = 3
- the y-intercept is 2(3²) + 2 = 20 or in coordinates of (0, 20)
<u>Question-2:</u>
How to make the graph of y = 2(x - 3)² + 2 through the transformation.
<u>The Process:</u>
To plot the graph of y = 2(x - 3) ² + 2 we apply for the following transformation order:
Step-1: clearly, to obtain the graph of y = (x - 3)² we shift the graph of y = x² to the right 3 units.
Step-2: to obtain the graph of y = 2(x - 3)², we stretch the graph of y = (x - 3)² by a factor of 2 (in other words, multiply each y-coordinate by 2).
Step-3: finally, to obtain the graph of y = 2(x - 3)² + 2 we shift the graph of y = 2(x - 3)² upward 3 units.
Thus the construction of the graph y = 2 (x - 3) ² + 2 is completed.
The graph of y = 2(x - 3) ² + 2 is drawn by the combination of shifting the graph of y = x² to the right 3 units and upward 2 units, and also stretch by a factor of 2. Between vertical shift and stretch steps, it is the same whatever steps are taken first.
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Notes
- The transformation of graphs is changing the shape and location of a graph.
- There are four types of transformation geometry: translation (or shifting), reflection, rotation, and dilation (or stretching/shrinking).
- In this case, the transformation is shifting horizontally and vertically and also stretching vertically.
In general, given the graph of y = f(x) and v > 0, we obtain the graph of:
by shifting the graph of
upward v units.
by shifting the graph of
That's the vertical shift, now the horizontal one. Given the graph of y = f(x) and h > 0, we obtain the graph of:
by shifting the graph of
by shifting the graph of
Hence, the combination of vertical and horizontal shifts is as follows:
The plus or minus sign follows the direction of the shift, i.e., up-down or left-right .
Notice the following definitions for stretch and shrink.
- In general, given the graph of
, we obtain the graph of
by stretching
or shrinking
the graph of
- In general, given the graph of
by stretching
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