standard_description: Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
problem_activity: Activity
The purpose of this activity is to strengthen connections between zeros of a function (input(s) when the output is 0), the output when the input is 0, and the coordinates of the intercept(s) of a graph of the function.
Launch
To ensure that students understand the task before beginning, display one of the functions for all to see, and test an input value that is
not
a zero of the function. For example, evaluate
\(a(\text-5)=\text-5 - 5\)
. Is the output 0? No, the output is -10. So -5 is not an input that would give an output of 0. Draw students’ attention to the list of possible inputs, and tell them that the task is to find the input or inputs for each function that gives an output of 0.
Student Facing
Here are the definitions of some functions, followed by some possible inputs for the functions.
\(a(x)=x - 5\)
\(b(x)=x + 5\)
\(c(x)=x-3\)
\(d(x)=x+1\)
\(f(x)=3x - 6\)
\(g(x)=3x + 6\)
\(h(x)=(x+5)(x+3)\)
\(m(x)=(x+1)(x-3)\)
\(n(x)=(3x-6)(x-5)\)
Possible inputs: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, and 5.
For each function, decide which input or inputs would give an output of 0.
Here are graphs of
\(b\)
,
\(f\)
, and
\(m\)
. Label each intercept with its coordinates, and be prepared to explain how you know.
[IMG]
Student Response
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Activity Synthesis
Display the correct responses to the first question and ask students to check their work. Invite a student to demonstrate how they know that
\(g(\text-2)=0\)
and
\(m(3)=0\)
(or choose a different function that any students struggled with). The important takeaway is not to look for shortcuts or patterns, but rather understand by evaluating the function why a given number is a zero of the function.
Invite students to share their responses and reasoning to the second question. If not mentioned, emphasize that:
The
\(y\)
-intercept of a graph corresponds to the output when the input is 0, so to find it, evaluate the function when
\(x\)
is 0. For example,
\(b(0)=5\)
, so the coordinates are
\((0,5)\)
of an intercept of the graph.
The
\(x\)
-intercepts correspond to the values found in the first question, the inputs that give an output of 0.
Any time you are dealing with an intercept, one of the coordinates must be 0. The coordinates of an
\(x\)
-intercept are
\((\text{something},0)\)
. The coordinates of a
\(y\)
-intercept are
\((0,\text{something})\)
.


aligns: no