sketch on a diagram the graphs of
1)y=x^2
2)y=(x+a)^2
3)y=b(x+a)^2
4)y=b(x+a)^2+c
To sketch the graphs of the given equations, you will need to understand the basic shape and transformations of the graph of a quadratic function.
1) y = x^2:
The graph of y = x^2 is a simple U-shaped curve known as a parabola. Its vertex is located at the origin (0, 0), which is the lowest point on the graph. As x increases or decreases, the value of y increases, resulting in the upward opening shape.
2) y = (x + a)^2:
To sketch the graph of y = (x + a)^2, you need to understand the effect of the translation of the function. The "a" value represents horizontal shift, meaning the graph of y = (x + a)^2 will shift "a" units to the left if a is negative or to the right if a is positive.
3) y = b(x + a)^2:
In this equation, the "b" value represents vertical stretch or compression. If b > 1, the graph will be stretched vertically, while if 0 < b < 1, the graph will be compressed vertically.
4) y = b(x + a)^2 + c:
This equation includes both a vertical shift and a horizontal shift, as well as vertical stretch or compression. The "c" value represents the vertical shift, moving the entire graph "c" units up if c is positive or down if c is negative.
To summarize the steps for sketching each equation:
1) For y = x^2, start at the vertex (0, 0) and draw a smooth U-shaped curve.
2) For y = (x + a)^2, shift the graph horizontally by "a" units. If a is negative, shift to the left; if a is positive, shift to the right.
3) For y = b(x + a)^2, apply vertical stretch or compression by adjusting the value of "b". Values greater than 1 stretch the graph, and values between 0 and 1 compress it.
4) For y = b(x + a)^2 + c, in addition to the previous steps, shift the graph vertically by "c" units. If c is positive, shift up; if c is negative, shift down.
By following these steps, you can sketch the graphs of the given equations. Remember to label any significant points, such as the vertex or any x- or y-intercepts, to further improve the accuracy of your sketch.