Lesson 03 · Where output becomes zero
Roots & X-Intercepts
Connect three languages for one event: a root is an x-value where the function equals zero, and the graph records it as an x-intercept.
The learner question
What does solving a quadratic look like on its graph?
Solving f(x) = 0 asks where the graph reaches height zero. Those x-values are roots or zeros; the corresponding points are x-intercepts.
Same event, three names
f(r) = 0
r is a root or zero. The graph contains the x-intercept (r, 0).
Worked model
Read y = (x − 1)(x + 3) from its factors.
A product is zero when at least one factor is zero. That gives two x-values where the parabola reaches the x-axis.
01Set the output to zero
0 = (x − 1)(x + 3)
Roots concern the places where y = 0.
02Use the zero-product property
x − 1 = 0 or x + 3 = 0
At least one factor must become zero.
03Name the intercepts
x = 1 or x = −3
The graph crosses at (1, 0) and (−3, 0).
vertex (-1, -4)
axis x = -1
Two distinct roots
crosses twice
The parabola passes through the x-axis at two different x-values.
One repeated root
touches once
The vertex lands on the x-axis, so the graph touches and turns.
No real roots
never reaches y = 0
The entire parabola stays above or below the x-axis.
Root geometry lab
Predict how many real roots the graph has.
vertex (0.5, -2.25)
axis x = 0.5
Current equation
y = (x − 2)(x + 1)
Do not confuse intercepts: the y-intercept occurs when x = 0. Roots occur when y = 0.
Check transferFresh cases, same underlying idea
Roots & X-Intercepts check1 / 3