standard_description: (+)Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
problem_activity: Warm-up
This warm-up highlights the three forms of quadratic expressions students have seen so far. It reinforces how each form gives different insights into the graph of a quadratic function.
Launch
Arrange students in groups of 2. Give students a minute of quiet think time, and then ask them to share their thinking with a partner.
Student Facing
Expressions in different forms can be used to define the same function. Here are three ways to define a function
\(f\)
.
\(f(x)=x^2-4x+3\)
(standard form)
\(f(x)=(x-3)(x-1)\)
(factored form)
\(f(x)=(x-2)^2-1\)
(vertex form)
Which form would you use if you want to find the following features of the graph of
\(f\)
? Be prepared to explain your reasoning.
the
\(x\)
-intercepts
the vertex
the
\(y\)
-intercept
Student Response
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Activity Synthesis
Invite students to share their responses and explanations. Encourage students to name specific features of the graph of
\(f\)
. If time permits, consider asking students to use all of the info they can gather to sketch a graph.
If not already mentioned by students, point out that while the standard form allows us to find the
\(y\)
-intercept more readily, the
\(y\)
-intercept can always be found by evaluating the function at
\(x=0\)
.


aligns: no