Translating Curves and Equations

Front

What is the effect of adding a constant to the equation of a curve, such as in y=|x-2|+4?

Back

Adding a constant to the equation of a curve shifts the curve vertically. In this case, the curve y=|x-2| is shifted up by 4 units.

Front

How does the equation y=|x|+2 differ from y=|x+2|+2 in terms of graph transformation?

Back

The equation y=|x|+2 shifts the graph of y=|x| up by 2 units, while y=|x+2|+2 shifts the graph left by 2 units and then up by 2 units.

Front

If the equation of a curve is y=|x-1|, what is the vertex of this curve?

Back

The vertex of the curve y=|x-1| is at the point (1, 0).

Front

What does the absolute value function represent in a graph?

Back

The absolute value function represents the distance of a number from zero on the number line, resulting in a V-shaped graph.

Front

How do you determine the y-value of a curve at a specific x-value, such as x=3 for the curve y=|x-2|?

Back

To find the y-value, substitute x=3 into the equation: y=|3-2| = |1| = 1.

Front

What is the general form of the equation for a vertically shifted absolute value function?

Back

The general form is y=|x-h|+k, where (h, k) is the vertex of the graph.

Front

What is the significance of the vertex in the graph of an absolute value function?

Back

The vertex is the point where the graph changes direction and is the minimum or maximum point of the function.

Front

How does the equation y=x differ from y=x+4 in terms of graph position?

Back

The equation y=x is a straight line through the origin, while y=x+4 is the same line shifted up by 4 units.

Front

What is the effect of changing the coefficient of x in the equation y=|mx|?

Back

Changing the coefficient m affects the slope of the lines in the V-shape of the absolute value function.

Front

What does the term 'translation' refer to in the context of graphing functions?

Back

Translation refers to shifting a graph horizontally or vertically without changing its shape.