Question

Which is the graph of y= 2(x-3)^2 +2

Answer 1

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 $\boxed{ \ y = a(x - h)^2 + k \ }$.

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 $\boxed{ \ y = ah^2 + k \ }$ 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:  

• $\boxed{ \ y = f(x) + v \ }$ by shifting the graph of $\boxed{ \ y = f(x) \ }$ upward v units.  
• $\boxed{ \ y = f(x) - v \ }$ by shifting the graph of $\boxed{ \ y = f(x) \ }$ 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:  

• $\boxed{ \ y = f(x + h) \ }$ by shifting the graph of $\boxed{ \ y = f(x) \ }$ to the left h units.  
• $\boxed{ \ y = f(x - h) \ }$ by shifting the graph of $\boxed{ \ y = f(x) \ }$ to the right h units.

Hence, the combination of vertical and horizontal shifts is as follows:  

$\boxed{ \ y = f(x \pm h) \pm v \ }$  

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 $\boxed{y = f(x)}$, we obtain the graph of $\boxed{y = cf(x)}$ by stretching $\boxed{ \ c > 1 \ }$ or shrinking $\boxed{ \ 0 < c < 1 \ }$ the graph of $\boxed{y = f(x)}$ vertically by a factor of c.
• In general, given the graph of $\boxed{y = f(x)}$, we obtain the graph of $\boxed{y = f(cx)}$ by stretching $\boxed{ \ 0 < c < 1 \ }$ or shrinking $\boxed{ \ c > 1 \ }$ the graph of $\boxed{y = f(x)}$ horizontally by a factor of c.

Learn more  

1. What is the y-intercept of the quadratic function  f(x) = (x – 6)(x – 2)? brainly.com/question/1332667
2. Transformations that change the graph of (f)x to the graph of g(x) brainly.com/question/2415963
3. Which statement correctly describes the graph  brainly.com/question/10929552

Answer 2

Your answer is linked in this picture attached: