In the following problems, we'll use visualizations with sliders (like the one below) to reason about lines and the equations that describe them.
In the visual above, can you imagine what would happen if you were able to keep moving the slider further and further to the right?
The slider below modifies one of the parameters of a linear equation. What could we reasonably expect to happen to the line if the slider were to move to a negative value?
These steepness (called the slope) of the two lines are controlled by the sliders below. What would happen to the lines if the sliders had the same value?
Hint: pay attention to the value labels on the sliders; even if the sliders are in the same position, they may not correspond to the same value.
In the equation \(y=kx\), any change to the value of \(x\) results in a change to the value of \(y\).
If adding \(3\) to \(x\) results in \(y\) growing by \(12\), what is the value of \(k\)?
The slider below controls the slope of all the lines. If the lines below are described by equations in the form \(y=mx+b\), which parameter is the slider controlling?
The equation that powers the visualization below is \(y=mx + b\). \(m\) controls the slope of the line and \(b\) controls its position. Is it possible to make every line shown above in the format \( y = mx + b ?\)
We want the line shown to pass through the marked red point at \( (2,2) .\) Which values of \(m\) will work? Mark all that apply and note that \(b\) can vary. \[ y = mx + b \]
In these last few questions, we've seen how the equation \(y=mx+b\) relates to the graph of a line and particularly how the parameter \(m\) describes its slope.
In the course ahead, we'll further investigate this equation and others, developing a deeper intuition about how changing one part of an equation affects the shape of its graph.