To find the relationship between the area of a parallelogram after dilation and the scale factor, we can use the properties of dilation. When a two-dimensional figure is dilated, the area of the figure is multiplied by the square of the scale factor.
Let's denote the original area of the parallelogram as A₁ and the area of the parallelogram after dilation as A₂. The scale factor required for dilating the parallelogram is denoted by y.
Given that the original area A₁ = 1 square centimeter, when we apply the dilation, the new area A₂ will be:
A₂ = A₁ × y²
We can now replace A₁ with 1 (since the original area is 1 square centimeter) and A₂ with x (which represents the new area in square units):
x = 1 × y²
x = y²
Now, solving for y, we get:
y² = x
y = √x
So the equation that gives us the scale factor y in terms of the area after dilation x is:
y = √x
We can plot this relationship on a graph with the x-axis representing the area after dilation (x) and the y-axis representing the scale factor (y). The graph will show how the scale factor changes as the area of the parallelogram changes after dilation.
To sketch a graph of y = √x:
- Start with the point (0,0), because when there is no area after dilation, the scale factor is zero.
- As x increases, y will also increase because the scale factor increases as the area increases.
- The shape of the graph will be a curve that starts at the origin and rises to the right. The curve gets less steep as x increases because the increase in the scale factor decreases relative to the area.
- Remember that y is always positive because we cannot have a negative scale factor, so the graph only exists in the first quadrant of the coordinate plane.
Here's how to sketch the graph:
1. Mark the x-axis as "Area after dilation (x square units)" and the y-axis as "Scale factor (y)".
2. Draw a curve that starts at the origin (0,0) and moves upward to the right. The curve will be a half-parabola shape opening to the right since y = √x is a square root function.
3. Make sure the curve gets shallower as x increases because the function y = √x increases at a decreasing rate for larger values of x.
4. You can label a few points on the graph to indicate specific areas and their corresponding scale factors, for example, (1, 1), (4, 2), (9, 3), and so on, because we know that squaring 1 gives 1, squaring 2 gives 4, and squaring 3 gives 9, reflecting the relationship y = √x.
That's the description of the graph for the equation y = √x, where y is the scale factor and x is the area after dilation of a parallelogram that originally had an area of 1 square centimeter.
