Explain why the graph of an equation in the form y=kx always goes through the origin. Give an example of a graph that shows direct variation and one that does not show direct variation.

I don't really get the question.

If y = kx, where k is any constant, y always equals 0 when x = 0. Therefore the origin (0,0) is on the line that is the graph of that function.

y = kx shows direct variation and y = k/x does not

y = xk

No problem! Let's break it down step by step.

First, let's understand the equation y = kx, where k is a constant. This equation represents a linear relationship between the variables x and y. In this specific form, it is called the equation of direct variation or direct proportionality.

To explain why the graph of y = kx always goes through the origin, let's consider what the equation represents. In this equation, k represents the constant of variation or proportionality. It indicates how the variables x and y are related to each other.

When x = 0, the equation reduces to y = k(0), which simplifies to y = 0. This means that when x is zero, y will always be zero. In other words, the origin (0,0) is a solution to this equation, and therefore the graph of y = kx passes through the origin.

Now let's consider an example of a graph that shows direct variation. Let's say we have the equation y = 3x. When we plot this equation on a graph, we will see that as x increases, y also increases proportionally. The points on the graph will form a straight line passing through the origin, with a slope of 3.

On the other hand, let's consider an example of a graph that does not show direct variation. For instance, if we have the equation y = x^2. When we plot this equation, we will observe a parabolic curve, not a straight line passing through the origin. This equation does not show direct variation because as x increases, y will not increase proportionally but will be determined by the square of x.

So, in summary, the graph of y = kx always goes through the origin because the equation represents a direct variation between x and y. An example of a graph that shows direct variation is y = 3x, while an example that does not show direct variation is y = x^2.

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