Exponential Function Properties

Algebra 1 · Axiom Academy

LESSON Exponential Function Properties How the two numbers in control where the curve starts and whether it climbs or fades toward zero. Every exponential function has the same skeleton: a starting amount a , repeatedly multiplied by a base b , x times. The whole thing hinges on one fact — when x=0 , the base term b^ 0 =1 , so the curve is pinned to the height a no matter what b is. That makes a the y-intercept , the initial value. at x=0 the curve always passes through a 2. Exponential Growth ( b > 1 ) When the base b is greater than 1 , each step multiplies the previous value by b — so the function doesn't add a fixed amount, it scales up again and again. Small at first, then startlingly steep: that compounding is exponential growth. Start at 1 , double each step: . Start at 3 , grow each step: . 3. Exponential Decay ( 0 < b < 1 ) When the base is a fraction between 0 and 1 , every step multiplies by less than one, so the value shrinks — fast at first, then slower. It heads toward the horizontal asymptote y=0 , getting arbitrarily close but never reaching it. Start at 1 , halve each step: . Start at 100 , lose each step: . 4. The Role of the Coefficient a The coefficient a vertically stretches the graph. Change a and the whole curve scales up or down by that factor — but the shape stays identical, because the base b (the growth rate) is untouched. Since the curve passes through (0,a) , changing a just slides the starting height.

This is the written version of the interactive lesson above. See the full Algebra 1 course.