4 mistakes learners make with harder graphs, each one named and explained.
A curve is a curve all the way along, and the places it never reaches matter as much as the places it goes through.
You draw y = 2 to the power x meeting the x axis on the left, and say it gets to zero eventually.
The curve gets so close that at the scale you are drawing it looks like it lands.
Question. Does the graph of y = 2 to the power x ever touch the x axis?
A common answer. yes, out to the left it reaches zero
The answer. no. It gets closer and closer but never touches
Why. Halving again and again makes a smaller number every time, and never zero.
An asymptote is a line the curve aims at and never reaches. Leave a gap.
You work out the y intercept of y = 2 to the power x as 0, because the power is 0.
A power of zero reads like zero lots of something, and zero lots of anything has always been zero.
Question. What is the value of y = 2 to the power x when x = 0?
A common answer. 0
The answer. 1
Why. Each step down divides by 2, so from 2 you reach 1, not 0.
Count down the powers: 8, 4, 2, then 1. Every exponential crosses at 1.
You make a table for y = 1/x, write 0 when x = 0, and plot a point there.
Zero is the answer that zero usually gives, and a table with a gap in it looks unfinished.
Question. What is the value of y = 1/x when x = 0?
A common answer. 0
The answer. there is no value. The curve never crosses that line
Why. Dividing by a smaller and smaller number gives a bigger and bigger answer, so there is no value at zero.
Leave the gap in the table. The gap is the asymptote.
You plot a quadratic correctly and then join the crosses with a ruler, so the bottom comes to a point.
Joining the points is what you did for every graph before this one, and a ruler feels more accurate than a free hand.
Question. You have plotted y = x squared. How should the points be joined?
A common answer. with a ruler, point to point
The answer. with a single smooth curve
Why. The curve exists between your points too, and a ruler puts corners where there are none.
If the equation has a power above one, put the ruler down.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.