Describe the following transformations on y=x^2:
y=-4(2x+10)^2-7
I really don’t understand transformations, please help
2x+10 = 2(x+5) so, you have
shift 5 left: y = (x+5)^2
shrink horizontally by a factor of 2: y = (2(x+5))^2 = (2x+10)^2
stretch vertically by a factor of 4: y = 4(2x+10)^2
reflect in x-axis: y = -4(2x+10)^2
shift down 7: y = -4(2x+10)^2-7
Sure! I'll be happy to explain the transformations that are happening in the equation y = -4(2x+10)^2 - 7.
To understand transformations, it's important to know the basics of graphing a simple quadratic function y = x^2. This is the parent function or the original graph.
Now, let's go through each transformation step-by-step:
1. Reflection in the x-axis: The negative sign in front of the equation, -4(2x+10)^2 - 7, reflects the graph over the x-axis. This means that all the y-values will be flipped to their negative counterparts.
2. Vertical Stretch/Compression: The 4 in -4(2x+10)^2 - 7 represents a vertical stretch or compression. Since the coefficient is greater than 1, it indicates a vertical compression. In other words, the graph will be narrower vertically. If the coefficient was less than 1, it would indicate a vertical stretch, making the graph wider vertically.
3. Horizontal Compression: The 2 in -4(2x+10)^2 - 7 represents a horizontal compression. It means that the graph will be narrower horizontally compared to the parent function (y = x^2) by a factor of 2.
4. Horizontal Translation: The -10 inside the parentheses, -4(2x+10)^2 - 7, represents a horizontal translation, also known as a shift. It indicates that the graph is moving 10 units to the left. If the number inside the parentheses were positive, it would indicate a rightward shift.
5. Vertical Translation: The -7 at the end of the equation, -4(2x+10)^2 - 7, represents a vertical translation, meaning the entire graph is moving 7 units downward. If the number were positive, it would move the graph upward.
To summarize, the given equation -4(2x+10)^2 - 7 represents a quadratic function that is reflected over the x-axis, vertically compressed, horizontally compressed, horizontally translated 10 units to the left, and vertically translated 7 units downward compared to the parent function y = x^2.
I hope this explanation helps!