23:T52a,This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). For g(x) = 2f(x-1) + 3, we apply transformations in this order: (1) f(x-1) shifts right 1 unit (since we have x minus 1), (2) multiplying by 2 gives 2f(x-1), which stretches vertically by factor 2, (3) adding 3 gives 2f(x-1) + 3, which shifts up 3 units. Choice B correctly identifies this sequence: shift right 1, stretch vertically by factor 2, then shift up 3. Choice A incorrectly states shift left instead of right, while C and D misidentify the transformation types entirely. The tricky part: horizontal shifts are opposite to the sign—f(x + 3) shifts LEFT 3 because you're subtracting 3 from x-coordinates, and f(x - 2) shifts RIGHT 2.24:T4c6,This question tests your understanding of how algebraic transformations of functions—like f(x) + k, k·f(x), f(kx), and f(x + k)—affect their graphs, and how to recognize even and odd functions from their symmetry properties. Function transformations come in four main types: (1) f(x) + k shifts the graph vertically (up if k > 0, down if k < 0), (2) k·f(x) stretches vertically if |k| > 1 or compresses if 0 < |k| < 1 (and reflects across x-axis if k < 0), (3) f(x + k) shifts horizontally—LEFT if k > 0, RIGHT if k < 0 (opposite of what you might expect!), (4) f(kx) compresses horizontally if |k| > 1 or stretches if 0 < |k| < 1 (and reflects across y-axis if k < 0). In g(x) = -f(x), we're multiplying all output values by -1, which reflects the graph across the x-axis—every point (x, y) becomes (x, -y). Choice A correctly identifies this as a reflection across the x-axis. Choice B incorrectly suggests y-axis reflection (that would be f(-x)), while C and D confuse reflections with shifts or compressions. Transformation memory aid: negative sign outside the function (-f(x)) flips y-values, creating x-axis reflection; negative sign inside the function (f(-x)) flips x-values, creating y-axis reflection.25:T612,

Transformations of Functions and Graphs Practice Test

Show the symmetry type by identifying whether $f(x)=x^3-x$ is even, odd, or neither.

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