What transformations of f(x) = x2 are applied to the function g(x) = 2(x-8)2+3
To find the transformations applied to the function g(x), we can compare it to the original function f(x) = x^2.
First, let's focus on the expression inside the parentheses: (x-8). This expression represents a horizontal translation of 8 units to the right. So, g(x) is shifted 8 units to the right compared to f(x).
Next, let's consider the expression (x-8)^2. This represents a vertical compression or stretching and a reflection. The coefficient 2 compresses the graph vertically by a factor of 2 compared to f(x). The square term represents a reflection about the y-axis.
Finally, the constant term 3 shifts the graph vertically upwards by 3 units compared to f(x).
In summary, the transformations applied to f(x) = x^2 to obtain g(x) = 2(x-8)^2+3 are:
1. Horizontal translation of 8 units to the right.
2. Vertical compression by a factor of 2.
3. Reflection about the y-axis.
4. Vertical shift upwards by 3 units.
To determine the transformations applied to the function g(x) = 2(x-8)^2+3, we need to compare it to the original function f(x) = x^2.
1. Horizontal translation: The function g(x) = 2(x-8)^2+3 is obtained by shifting the graph of f(x) = x^2 eight units to the right. This is because the term (x-8) in the function represents a horizontal shift of 8 units to the right.
2. Vertical translation: The "+3" term in the function g(x) = 2(x-8)^2+3 translates the graph of f(x) = x^2 vertically upwards by 3 units. This is because the "+3" added to the function represents a vertical shift of 3 units upwards.
3. Vertical stretch: The "2" coefficient multiplied by (x-8)^2 in g(x) = 2(x-8)^2+3 indicates that the graph of f(x) = x^2 is vertically stretched by a factor of 2. This means that the graph is narrower compared to the original function f(x) = x^2.
To summarize, the transformations applied to the function g(x) = 2(x-8)^2+3 are a horizontal translation of 8 units to the right, a vertical translation of 3 units upwards, and a vertical stretch by a factor of 2.
To determine the transformations applied to the function g(x) based on the function f(x), you need to rewrite g(x) in a similar form to f(x).
Let's break it down:
g(x) = 2(x-8)^2 + 3
Comparing it with f(x) = x^2, we can identify the following transformations:
1. Horizontal Translation:
In f(x), the vertex (minimum point) is located at (0,0). However, in g(x), the vertex is located at (8,3). This means that g(x) has been horizontally shifted 8 units to the right.
2. Vertical Translation:
In f(x), the value of f(x) is 0 when x = 0. In g(x), the value is 3 when x = 8. This indicates a vertical translation 3 units upward.
3. Vertical Stretch:
The coefficient 2 in front of (x-8)^2 implies a vertical stretch. Compared to f(x), g(x) is stretched vertically by a factor of 2.
Now, let's summarize the transformations:
1. Horizontal Translation: The graph has been shifted horizontally 8 units to the right.
2. Vertical Translation: The graph has been shifted vertically 3 units upward.
3. Vertical Stretch: The graph has been stretched vertically by a factor of 2.