Graphs and Transformations Flashcards
What is the basic shape of y = x²?
A parabola opening upwards.
What is the shape of y = x³?
An S-shaped curve increasing from left to right.
What does the graph of y = √x look like?
Starts at (0,0) and curves slowly upwards.
What is the graph of y = 1/x?
A hyperbola with vertical and horizontal asymptotes.
What is the effect of y = f(x) + a?
Vertical translation up by a units.
What is the effect of y = f(x + a)?
Horizontal translation left by a units.
What is the effect of y = af(x)?
Vertical stretch by a factor of a.
What is the effect of y = f(ax)?
Horizontal stretch by a factor of 1/a.
What is the effect of y = -f(x)?
Reflection in the x-axis.
What is the effect of y = f(-x)?
Reflection in the y-axis.
What is the composite transformation of y = -f(-x)?
Reflection in both axes; rotation of 180° about the origin.
What does the graph of y = |f(x)| look like?
All parts of the graph below the x-axis are reflected above it.
What does the graph of y = f(|x|) look like?
The graph for x ≥ 0 is reflected across the y-axis.
How do you find where two graphs intersect?
Set their equations equal and solve for x.
What does the modulus function y = |x| look like?
V-shape with vertex at the origin.
How do you sketch the graph of a translated function?
Apply the transformation to key points and replot.
What is the asymptote of y = 1/x?
Vertical at x = 0 and horizontal at y = 0.
What is the asymptote of y = 1/(x - a)?
Vertical at x = a and horizontal at y = 0.
How do transformations affect asymptotes?
Vertical shifts affect horizontal asymptotes; horizontal shifts affect vertical asymptotes.
What is the importance of understanding transformations?
They allow sketching of complex graphs from basic ones.
