Changing the Vertex
5 min
Narrative
In this Warm-up, students revisit quadratic expressions in different forms and what the expressions reveal about the graphs that represent them. They match equations and graphs representing two quadratic functions. The axes of the graphs are unlabeled, so students need to reason abstractly about the parameters in the expressions and the features of the graphs.
Student Task
Here are graphs representing two functions, f and g, given by f(x)=x(x+6) and g(x)=x(x+6)+4.
- Which graph represents each function? Explain how you know.
- Where does the graph of f meet the x-axis? Explain how you know.
Sample Response
- The lower graph (which goes through the origin) is the graph of f, and the upper one is the graph of g. Sample explanations:
- The value of g is always 4 greater than the value of f no matter which value of x we choose.
- Rewriting the equations in standard form gives f(x)=x2+6x and g(x)=x2+6x+4. This form shows that the y-intercept of the graph representing f is (0,0), and the y-intercept of the graph of g is (0,4).
- The expression that defines function f is in factored form. It shows that the two x-intercepts are (0,0) and (-6,0).
- x=-6 and x=0, because these are the two values of x that make x(x+6) equal to 0.
Activity Synthesis (Teacher Notes)
Invite students to share their matches and explanations. If one or more explanations in the Student Response are not mentioned, bring them up so that students can see multiple ways of reasoning about the equations and graphs.
Standards
- F-IF.C·Analyze functions using different representations
- HSF-IF.C·Analyze functions using different representations.
15 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Changing the Vertex
5 min
Narrative
In this Warm-up, students revisit quadratic expressions in different forms and what the expressions reveal about the graphs that represent them. They match equations and graphs representing two quadratic functions. The axes of the graphs are unlabeled, so students need to reason abstractly about the parameters in the expressions and the features of the graphs.
Student Task
Here are graphs representing two functions, f and g, given by f(x)=x(x+6) and g(x)=x(x+6)+4.
- Which graph represents each function? Explain how you know.
- Where does the graph of f meet the x-axis? Explain how you know.
Sample Response
- The lower graph (which goes through the origin) is the graph of f, and the upper one is the graph of g. Sample explanations:
- The value of g is always 4 greater than the value of f no matter which value of x we choose.
- Rewriting the equations in standard form gives f(x)=x2+6x and g(x)=x2+6x+4. This form shows that the y-intercept of the graph representing f is (0,0), and the y-intercept of the graph of g is (0,4).
- The expression that defines function f is in factored form. It shows that the two x-intercepts are (0,0) and (-6,0).
- x=-6 and x=0, because these are the two values of x that make x(x+6) equal to 0.
Activity Synthesis (Teacher Notes)
Invite students to share their matches and explanations. If one or more explanations in the Student Response are not mentioned, bring them up so that students can see multiple ways of reasoning about the equations and graphs.
Standards
- F-IF.C·Analyze functions using different representations
- HSF-IF.C·Analyze functions using different representations.