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.
Further explanation
The Problem:
Which is the graph of y = 2(x - 3)² + 2?
Question-1:
How to make a graph y = 2 (x - 3) ² + 2 through the concept of a parabola.
The Process:
The equation of a parabola is given by [tex]\boxed{ \ y = a(x - h)^2 + k \ }[/tex].
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 [tex]\boxed{ \ y = ah^2 + k \ }[/tex] 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)
Question-2:
How to make the graph of y = 2(x - 3)² + 2 through the transformation.
The Process:
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:
- [tex]\boxed{ \ y = f(x) + v \ }[/tex] by shifting the graph of [tex]\boxed{ \ y = f(x) \ }[/tex] upward v units.
- [tex]\boxed{ \ y = f(x) - v \ }[/tex] by shifting the graph of [tex]\boxed{ \ y = f(x) \ }[/tex] downward v units.
That's the vertical shift, now the horizontal one. Given the graph of y = f(x) and h > 0, we obtain the graph of:
- [tex]\boxed{ \ y = f(x + h) \ }[/tex] by shifting the graph of [tex]\boxed{ \ y = f(x) \ }[/tex] to the left h units.
- [tex]\boxed{ \ y = f(x - h) \ }[/tex] by shifting the graph of [tex]\boxed{ \ y = f(x) \ }[/tex] to the right h units.
Hence, the combination of vertical and horizontal shifts is as follows:
[tex]\boxed{ \ y = f(x \pm h) \pm v \ }[/tex]
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 [tex]\boxed{y = f(x)}[/tex], we obtain the graph of [tex]\boxed{y = cf(x)}[/tex] by stretching [tex]\boxed{ \ c > 1 \ }[/tex] or shrinking [tex]\boxed{ \ 0 < c < 1 \ }[/tex] the graph of [tex]\boxed{y = f(x)}[/tex] vertically by a factor of c.
- In general, given the graph of [tex]\boxed{y = f(x)}[/tex], we obtain the graph of [tex]\boxed{y = f(cx)}[/tex] by stretching [tex]\boxed{ \ 0 < c < 1 \ }[/tex] or shrinking [tex]\boxed{ \ c > 1 \ }[/tex] the graph of [tex]\boxed{y = f(x)}[/tex] horizontally by a factor of c.
Learn more
- What is the y-intercept of the quadratic function f(x) = (x – 6)(x – 2)? https://brainly.com/question/1332667
- Transformations that change the graph of (f)x to the graph of g(x) https://brainly.com/question/2415963
- Which statement correctly describes the graph https://brainly.com/question/10929552
