Appendix D Graphing Techniques Objective SWBAT review graphing
Appendix D Graphing Techniques Objective: SWBAT review graphing techniques of stretching & shrinking, reflecting, symmetry and translations.
Types of Graphs: 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
Identity Function Domain: all reals Range: all reals
Square Function (a quadratic formula) Domain: all reals Range: [0, ∞)
Cube Function Domain: all reals Range: all reals
Square Root Function Domain: [0, ∞) Range: [0, ∞)
Cube Root Function Domain: all reals Range: all reals
Absolute Value Function Domain: all reals Range: [0, ∞)
Vertical Shift 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.
Vertical Shift
Horizontal Shift 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|.
Horizontal Shift
Horizontal & Vertical Shifts If the shift happens outside the function, it is vertical. If the shift happens inside the function, it is horizontal.
Reflection, Stretches and Shrinks 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|.
Reflection, Stretches and Shrinks
Reflection, Stretches and Shrinks
Reflections �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 xaxis, 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.
Testing for Symmetry with Respect to an axis 1. In y = x 2 +4 replace x with –x 2. In x = y 2 - 3 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 1800 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
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