Graphs of Basic Functions

Pre-Calculus · Axiom Academy

LESSON Graphs of Basic Functions Read a parent function straight off its graph — its shape, its symmetry, and the part of the plane it can reach. 1. A Graph Is the Equation, One Point at a Time The graph of a function is nothing mysterious: for each input x , plot the point . Sweep x across the number line, drop a point at every height, and the whole curve appears. Below we build the quadratic parent f(x)=x^2 exactly this way. Each input x pairs with the output x^2 Replace x with -x and watch what the graph does. If the rule is unchanged, the left and right halves are mirror images across the y -axis — an even function. If the output flips sign instead, the graph looks the same after a half-turn about the origin — an odd function. f(-x)=f(x) . The point (x,y) has a partner (-x,y) . Parents x^2 and |x| are even. f(-x)=-f(x) . The point (x,y) has a partner (-x,-y) . Parents x , x^3 , and 1/x are odd. 3. Where a Function Lives: Domain & Range The domain is the set of inputs the rule allows — the graph s shadow on the x -axis. The range is the set of outputs it reaches — its shadow on the y -axis. Two parents make the idea concrete: the square root can t take negatives, and the reciprocal can t touch zero in either direction. Domain : negatives have no real square root. Range : the output is never negative. Domain : dividing by zero is undefined. Range : the output never equals zero. Both axes are asymptotes.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.