y = (x - 4)² + 4
or y = x² - 8x + 20
Further explanation
Transformation of a graph 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).
- In this case, the transformation is shifting horizontally or vertically.
- Translation (or shifting): moving a graph on an analytic plane without changing its shape.
- Vertical shift: moving a graph upwards or downwards without changing its shape.
- Horizontal shift: moving a graph to the left or right downwards without changing its shape.
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:
\(\boxed{\boxed{ \ y = f(x \pm h) \pm v \ }}\)
The plus or minus sign follows the direction of the shift, i.e., up-down or left-right
Given: \(\boxed{ \ g(x) = x^2 \ becomes \ f(x) = ? \ }\)
In the graph, notice the shifting of the vertex from (0, 0) to (4, 4).
From this, we can describe that from g(x) to f(x) there has been a shift to the right 4 units and upward 4 units.
Let us construct f(x) from g(x).
\(\boxed{ \ g(x) = y = x^2 \ } \rightarrow \boxed{ \ f(x) = y = (x + h)^2 + v \ }\)
We set h = -4 and v = +4 and we get the equation f(x) as
\(\boxed{\boxed{ \ f(x) = (x - 4)^2 + 4 \ }}\)
Let's expand it if we want to represent a standard form of a quadratic function, like this:
\(\boxed{ \ f(x) = x^2 - 8x + 16 + 4 \ }\)
\(\boxed{\boxed{ \ f(x) = x^2 - 8x + 20 \ }}\)
Conclusion
The graph of f(x) is drawn by the combination of shifting the graph of g(x) to the right 4 units and upward 4 units.
Learn more
- Transformations that change the graph of (f)x to the graph of g(x) https://brainly.com/question/2415963
- The similar problem https://brainly.com/question/1369568
- Determine the coordinates of the image of a point after the triangle is rotated 270° about the origin https://brainly.com/question/7437053
Keywords: transformations, the graph of f(x), resembles, g(x) = x², f(x) = (x - 4)² + 4, y = x² - 8x + 20, translation, shifting, right, upward
, horizontal, vertical
