Question

A line has equation \(y=x\) (a) Find the new equation if the line is shifted vertically up by 5 units. (b) Find the new equation if the line is shifted horizontally to the left by 5 units. (c) Compare your answers to (a) and (b) and explain your result graphically.

Short answer

Answer: The equation of the line after being shifted vertically up by 5 units and horizontally to the left by 5 units is \(y = x + 5\).

Step-by-step solution

Part (a): Shifting Line Vertically Up by 5 Units

To shift a line vertically, we need to add or subtract the required units from the y-coordinate of the equation of the line. In this case, we need to shift the line up by 5 units, so we will add 5 to the y-coordinate in the given equation. Given equation: \(y = x\) New equation after shifting up by 5 units: \(y - 5 = x \Rightarrow y = x+5\)

Part (b): Shifting Line Horizontally to the Left by 5 Units

To shift a line horizontally, we need to add or subtract the required units from the x-coordinate of the equation of the line. In this case, we need to shift the line to the left by 5 units, so we will add 5 to the x-coordinate in the given equation. Given equation: \(y = x\) New equation after shifting to the left by 5 units: \(y = x + 5\)

Part (c): Comparison of (a) and (b) and Graphical Explanation

After solving both parts (a) and (b), we have the following results: (a) When the line is shifted vertically up by 5 units: \(y = x + 5\) (b) When the line is shifted horizontally to the left by 5 units: \(y = x + 5\) Both of these transformations result in the same equation, \(y = x + 5\). Graphically, this indicates that shifting the line \(y = x\) up by 5 units or to the left by 5 units creates the same resultant line. This is true, because the line \(y = x\) has a slope of \(45^{\circ}\), so translating it up by 5 units creates the same result as translating it to the left by 5 units. The same line is found under both transformations.

Key concepts

Vertical Translation

When we talk about a vertical translation in the context of linear equations, we mean moving a graph up or down in relation to the y-axis. This happens when you change the y-coordinate values. Imagine lifting the whole line without changing its orientation.
This is simply adding or subtracting a fixed number from the whole equation. For the line described by the equation \(y = x\), translating it vertically by 5 units upward changes it to \(y = x + 5\).

Here's why this is intuitive:

• The equation \(y = x\) tells us that for every unit increase in \(x\), \(y\) also increases by one because they are equal.
• By adding 5 to \(y\), every point on the line is moved 5 steps up along the y-axis, essentially "lifting" the line.

This results in the same slope, meaning the angle it makes with the x-axis remains unchanged. Simply put, vertical translations do not affect the line's steepness, just its starting point on the graph.

Horizontal Translation

Horizontal translation involves moving the line side to side - from left to right or right to left - in relation to the x-axis. It's about shifting the complete graph without altering its tilt or steepness. When moving a line horizontally, we modify the x-coordinate's values.
For the equation \(y = x\), if we translate the line to the left by 5 units, the new equation becomes \(y = (x + 5)\).

Here is how horizontal translation works and why it results in the same output as vertical translation for a line like \(y = x\):

• To shift left, we add to the x-coordinate in the equation, which might seem counterintuitive, but keep in mind: making \(x + 5\) instead of just \(x\) means we need a smaller x to get the same y.
• This effectively moves every point of the line to the left by 5 units.

For specific lines, like \(y = x\), a horizontal translation results in the same output equation as a vertical translation does. This is due to their identical slopes, causing certain directional shifts to coincide numerically.

Graphical Representation

Graphical representation helps us visualize the effects of both vertical and horizontal translations on the line \(y = x\). The graph of a straight line can be manipulated by these translations without altering its slope, allowing it to maintain its angle with respect to the axes.

Imagine drawing the line \(y = x\), which forms a perfect diagonal that goes through the origin at \(45^{\circ}\). A vertical shift upward by 5 units takes every point and raises it, while a horizontal shift left by 5 units slides every point horizontally, leftward. Amazingly, the resulting line is identical in both cases, showing as \(y = x + 5\).

• A vertical translation changes the intercept with the y-axis, shifting the whole graph up or down but keeping the slope constant.
• A horizontal translation changes where the graph crosses the x-axis but similarly keeps the slope unchanged.

The graphical portrayal of these translations confirms that the line remains parallel to its initial position, emphasizing that such transformations mainly affect position, not inclination. Interactive graphs can be particularly useful in visualizing these concepts. Consider playing around with graphing tools to watch these translations in action.