Direct and Inverse Proportion

Summary

This video explains various types of proportional relationships in mathematics, including direct, inverse, and squared proportionality. It covers the principles behind equations like y = kx, y = k/x, and y = kx², illustrating how to calculate the constant of proportionality (k) and solve for unknown values using real-world examples. The video also explores how these relationships manifest on graphs and provides step-by-step problem-solving methods, making it an invaluable resource for understanding proportionality in algebra.

Takeaways

• 😀 Direct proportionality means that one variable (y) is directly related to another variable (x) by a constant factor (k), expressed as y = kx.
• 😀 A direct proportionality relationship produces a straight line on a graph that passes through the origin (0,0). The slope of this line represents the constant k.
• 😀 In direct proportionality, the formula y = kx can be used to calculate values when given one variable (x or y) and the other variable's corresponding value.
• 😀 The symbol for direct proportionality (y ∝ x) means the same as y = kx, but it's a shorthand notation.
• 😀 To find the constant of proportionality (k), substitute known values of y and x into the formula and solve for k.
• 😀 If y is directly proportional to x squared (y = kx²), the graph will be a parabola that passes through the origin.
• 😀 In inverse proportionality (y = k/x), y is inversely related to x, meaning as one variable increases, the other decreases.
• 😀 Inverse proportionality produces a hyperbolic graph, which approaches the axes but never touches them.
• 😀 To find k in inverse proportionality, use known values of y and x to substitute into y = k/x and solve for k.
• 😀 The formula for inverse proportionality (y = k/x) allows you to calculate values of y or x by rearranging the equation based on given variables.
• 😀 When y is inversely proportional to x squared (y = k/x²), the graph is similar to a 1/x graph but always remains above the x-axis due to the squaring of x.
• 😀 The method to find k and solve for unknown values is the same in both direct and inverse proportionality relationships: substitute known values, solve for k, and then use the formula to find the unknown variable.

Q & A

What does it mean when y is directly proportional to x?

It means that y equals a constant k multiplied by x, or y = kx. The graph of this relationship is a straight line that passes through the origin.

How do you find the constant of proportionality, k, when y is directly proportional to x?

You can find k by substituting known values for y and x into the equation y = kx, and then solving for k. For example, if y = 16 when x = 5, then k = 16 / 5 = 3.2.

How is directly proportional written symbolically?

Directly proportional is often written as y ∝ x, where the symbol '∝' represents proportionality.

What does the graph of y = kx look like?

The graph is a straight line that passes through the origin (0, 0) with a gradient equal to k.

How would you solve for x if y = 32 and y = 3.2x?

You substitute y = 32 into the equation, so 32 = 3.2x. Divide both sides by 3.2 to find x = 32 / 3.2 = 10.

How do you handle a situation where y is directly proportional to x squared?

The equation becomes y = kx². You can find k by using given values of y and x, and then use the formula to solve for other values of y or x.

What happens when y is inversely proportional to x?

In this case, y = k / x, where k is a constant. The graph of this relationship is a curve that approaches zero as x increases.

How do you calculate the constant k for an inverse proportional relationship?

Substitute the known values for y and x into y = k / x and solve for k. For example, if y = 12 and x = 3, then k = 12 * 3 = 36.

What does the graph of an inverse proportional relationship look like?

The graph of y = k / x is a hyperbola, approaching zero as x increases, and is located in the first and third quadrants.

How do you find the value of y when x = 2, given that y = 36 / x?

Substitute x = 2 into the equation y = 36 / x. Thus, y = 36 / 2 = 18.