Curriculum / Math / 9th Grade / Unit 8: Quadratic Equations and Applications / Lesson 12
Math
Unit 8
9th Grade
Lesson 12 of 15
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Write and analyze quadratic functions for geometric area applications.
The core standards covered in this lesson
A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
F.IF.C.8.A — Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
The foundational standards covered in this lesson
A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
A.SSE.B.3.A — Factor a quadratic expression to reveal the zeros of the function it defines.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
A rectangular garden has a width of $$x$$ feet. The length of the garden is $$4$$ feet longer than the width. Around the garden there is a pathway that measures $$3$$ feet across the path.
The expression below represents the area of the pathway, not including the garden.
$$(x+6)(x+10)-x(x+4)$$
Explain what each part of the expression represents in context of the situation.
a. $$x$$
b. $$x+4$$
c. $$x+10$$
d. $$x(x+4)$$
e. $$(x+6)(x+10)$$
An artist has a strip of wood $$8$$ feet long to make a large rectangular picture frame. The possible dimensions of the frame that the artist can create can be represented as $$x$$ and $$4-x$$, as seen in the diagram below.
What is the largest area the artist can frame with the $$8$$ feet of wood?
A set of suggested resources or problem types that teachers can turn into a problem set
Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
The area of a rectangular storage room in square feet is given by the function $${ A(x)=-2x^2+48x}$$, where $$x$$ represents the length of the room.Â
a. Which inequality represents all of the possible lengths, in feet, of the storage room?
i.  $$0<x<12$$
ii. $$12<x<24$$
iii. $$0<x<24$$
iv.  $$x<0 \space \mathrm{or} \space x>24$$
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b. What length of the storage room will maximize the area of the room? What is the maximum area?Â
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
Lesson 11
Lesson 13
Topic A: Deriving the Quadratic Formula
Describe features of the vertex form of a quadratic function and write quadratic equations in vertex form from graphs.
A.SSE.B.3 F.IF.B.4 F.IF.C.8
Complete the square.
A.SSE.B.3.B
Complete the square to identify the vertex and solve for the roots of a quadratic function.
A.REI.B.4.B A.SSE.B.3.B
Solve and interpret quadratic applications using the vertex form of the equation.
A.SSE.B.3.B F.IF.C.8.A
Convert and compare quadratic functions in standard form, vertex form, and intercept form.
F.IF.B.4 F.IF.C.9
Derive the quadratic formula. Use the quadratic formula to find the roots of a quadratic function.
A.REI.B.4.A
Determine the number of real roots of a quadratic function using the discriminant of the quadratic formula.
A.REI.B.4.B F.IF.C.7.A
Graph quadratic functions from all three forms of a quadratic equation.
F.IF.C.7.A
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Topic B: Transformations and Applications
Describe transformations to quadratic functions. Write equations for transformed quadratic functions.
F.BF.B.3
Graph and describe transformations to quadratic functions in mathematical and real-world situations.
Write and analyze quadratic functions for projectile motion and falling bodies applications.
A.CED.A.2 F.IF.C.8.A F.IF.C.9
A.CED.A.2 F.IF.C.8.A
Write and analyze quadratic functions for revenue applications.
A.CED.A.2 F.BF.A.1.B F.IF.C.8.A
Solve and identify solutions to systems of quadratic and linear equations when two solutions are present.
A.REI.C.7 A.REI.D.11
Solve and identify solutions to systems of quadratic and linear equations when two, one, or no solutions are present.
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