Objective: SWBAT review graphing techniques of stretching & shrinking, reflecting, symmetry and translations.

Sketch the following different types of graphs in your notebook. Graph paper is on the side table. Notice the function and what the graph looks like

Domain: all reals Range: all reals

Domain: all reals Range: [0, ∞)

Domain: all reals Range: all reals

Domain: [0, ∞) Range: [0, ∞)

Domain: all reals Range: all reals

Domain: all reals Range: [0, ∞)

The graph of y = f(x) + k can be obtained from the graph of y = f(x) by vertically translating (shifting) the graph of the latter upward k units if k is positive and downward |k| units if k is negative. Graph y = |x|, y = |x| + 4, and y = |x| – 5.

The graph of y = f(x + h) can be obtained from the graph of y = f(x) by horizontally translating (shifting) the graph of the latter h units to the left if h is positive and |h| units to the right if h is negative. Graph y = |x|, y = |x + 4|, and y = |x – 5|.

If the shift happens outside the function, it is vertical. If the shift happens inside the function, it is horizontal.

The graph of g(x) = a * f(x) has the same general shape as the graph of f(x). If a > 1, the result is a vertical stretch (narrower) compared to the graph of f(x). If 0 < a < 1, the result is a vertical shrink (wider) compared to the graph of f(x). If a = –1, the result is a reflection in the x axis. Graph y = |x|, y = 2|x|, y = 0.5|x|, and y = –2|x|.

h(x)= - f(x) reflects across the x axis. h(x)= f(-x) reflects across the y axis. If it happens outside the function, reflects across the x axis. If it happens inside the function, reflects across the y axis.

Symmetry The graph of f (x) is cut in half by the y-axis with each half the mirror image of the other half. A graph with this property is said to be symmetric with respect to the y-axis. As this graph suggests, a graph is symmetric with respect to the y-axis if the point (-x, y) is on the graph whenever the point (x, y) is on the graph.

Symmetry Similarly, if the graph of g (x)were folded in half along the x-axis, the portion at the top would exactly match the portion at the bottom. Such a graph is symmetric with respect to the x-axis: the point (x, -y) is on the graph whenever the point (x, y) is on the graph.

1. In y = x 2 +4 replace x with –x 2. In x = y replace y with –y 3. In x 2 +y 2 = 16 substitute –x for x and –y for y

Symmetry Another kind of symmetry occurs when a graph can be rotated around the origin, with the result coinciding exactly with the original graph. Symmetry of this type is called symmetry with respect to the origin. A graph is symmetric with respect to the origin if the point (-x, -y) is on the graph whenever the point (x, y) is on the graph.

Testing for Symmetry with Respect to the Origin Are the following graphs symmetric with respect to the origin?

Testing for Symmetry with Respect to the Origin Are the following graphs symmetric with respect to the origin?

Tests for Symmetry