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2.3 Stretching, Shrinking, and Reflecting Graphs Quiz What’s the transformation of y = cf(x) if c>1 from y = f(x)? A vertical shrinking B vertical stretching C horizontal shrinking D horizontal stretching Vertical Stretching Discussion y f(x) = x2 f(x) = 2x2 f(x) = 3x2 f(x) = 4x2 Notice: the x intercept doesn’t change. x Vertical Stretching If a point (x,y) lies on the graph of y = f(x), then the point (x, cy) lies on the graph of y = cf(x). If c>1, then the graph of y = cf(x) is a vertical stretching of the graph of y = f(x) by applying a factor of c. Vertical Shrinking Discussion y f(x) = x2 f(x) = ⅟2x2 f(x) = ⅟3x2 f(x) = ⅟4x2 Notice: The xintercept doesn’t change, either. x Vertical Shrinking If a point (x,y) lies on the graph of y = f(x), then the point (x,cy) lies on the graph of y = cf(x). If 0 < c < 1, then the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x) by applying a factor of c. Vertical Stretching and Shrinking y c>1 y = cf(x) 0<c<1 y = cf(x) x 0<c<1 y = cf(x) Points on the x-axis doesn’t change c>1 y = cf(x) Horizontal Stretching and Shrinking Discussion y f(x) = |x|-1 f(x) = |⅟2x|-1 f(x) = |2x|-1 x Notice: the yintercept doesn’t change. Horizontal Stretching and Shrinking If a point (x,y) lies on the graph of y = f(x), then the point (x/c, y) lies on the graph of y = f(cx) If 0 < c < 1, then the graph of y = f(cx) is a horizontal stretching of the graph of y = f(x). If c > 1, then the graph of y = f(cx) is a horizontal shrinking of the graph of y = f(x). Horizontal Stretching and Shrinking y 0<c<1 y = f(cx) c>1 y = f(cx) c>1 y = f(cx) 0<c<1 y = f(cx) x Points on the y-axis doesn’t change Reflection y f(x) = x-3 g(x) = -x-3 g(x) f(x) h(x) = -x+3 x g(x) = f(-x) h(x) = -f(x) h(x) Reflection The graph of y = -f(x) is a reflection of the graph of f across the x-axis. The graph of y = f(-x) is a reflection of the graph of f across the y-axis y y = f(-x) x y = -f(x) Combining Transformations of graphs Example 1 y = - ⅟2|x - 4| + 3 Step 1: Identify the basic function. Step 2: shift right by 4 units. Step 3: vertical shrink by ½. Step 4: reflection across y-axis. Step 5: shift up by 3 units. f(x) = |x| f(x) = |x-4| f(x) = ½|x-4| f(x) = -½|x-4| f(x) = -½|x-4|+3 Homework PG. 111: 24-36(M3), 38, 41, 48, 53, 58, 63, 64, 79, 90 KEY: 18, 27, 49, 51 No Reading, next class is review, so no quiz! But bring your student handbook!