describe the transformations that were done to the parent function F(x)=X^2 to create g(x)=-2(x-6)^2 +5. Use complete sentences
f(x-6) ... shift right 6
2f(x-6) ... stretch vertically by 2
-2f(x-6) ... reflect across x-axis
-2f(x-6)+5 ... shift up 5
Well, it looks like some funny business has been done to the parent function F(x)=x^2 to create g(x)=-2(x-6)^2+5! Let's break it down.
First, we notice that a negative sign has been added in front of the whole function. That's quite the transformation! This "flips" the graph of the parent function upside down, giving it a new trick.
Next, we see that the "x" inside the function has been replaced with "x-6". This means the whole graph has been shifted 6 units to the right. It's as if the function decided to take a little stroll, six steps to the right!
Lastly, we have the factor of 2 in front of the squared term. This makes the graph get stretched vertically. It's like the function went to the gym and got a good workout, pumping some iron!
So, to summarize, the parent function F(x)=x^2 has undergone a flip, a stroll to the right, and a vertical stretch to become the new and hilarious function g(x)=-2(x-6)^2+5!
To create the function g(x) = -2(x-6)^2 + 5 from the parent function f(x) = x^2, several transformations have been applied.
1. Horizontal Translation: The function has been horizontally translated by 6 units to the right. This means that the vertex of the parabola has moved from the origin (0, 0) to the point (6,0).
2. Vertical Reflection: The function has been vertically reflected by a factor of -2. This means that the parabola has been flipped upside down, changing the direction it opens.
3. Vertical Compression: The function has been vertically compressed by a factor of 2. This means that the parabola has become narrower compared to the parent function. The coefficient of -2 in front of the equation indicates this compression.
4. Vertical Shift: The function has been vertically shifted upwards by 5 units. This means that the entire parabola has moved vertically upwards, shifting the vertex to the point (6, 5).
Overall, these transformations have transformed the parent function f(x) = x^2 into the function g(x) = -2(x-6)^2 + 5, resulting in a narrower, vertically reflected, horizontally translated, and vertically shifted parabola.
To transform the parent function f(x) = x^2 into the function g(x) = -2(x-6)^2 + 5, the following transformations were applied:
1. Horizontal Translation: The function has been horizontally translated to the right by 6 units. This is evident from the term (x-6), where the original x-values are reduced by 6 units.
2. Vertical Reflection: The function has been vertically reflected or flipped upside down. This is apparent from the coefficient -2, which flips the original parabola upside down.
3. Vertical Compression: The function has been vertically compressed. The coefficient 2 in front of the parent function x^2 results in a steeper, narrower parabola compared to the original function.
4. Vertical Translation: The function has been vertically translated upwards by 5 units. This can be observed from the constant term +5, where the entire graph of the transformed function g(x) has been shifted upward by 5 units.
Therefore, the transformations applied to the parent function f(x) = x^2 to create g(x) = -2(x-6)^2 + 5 are horizontal translation to the right by 6 units, vertical reflection, vertical compression, and vertical translation upwards by 5 units.