Instructions
You are helping a community garden plan a new rectangular planting area. Use algebraic models to make decisions, interpret graphs, and explain your reasoning.
- Show your work for every calculation.
- Use exact answers when possible. Round decimal answers to the nearest tenth unless instructed otherwise.
- Use a calculator only when a question requires approximation.
- Complete the core questions first. The extension questions are optional.
- Check that your answers make sense in the real-world situation.
Learning goals:
- Factor and solve quadratic equations.
- Use the quadratic formula and identify extraneous or unreasonable solutions.
- Interpret the vertex, intercepts, and axis of symmetry of a quadratic function.
- Model a real-world situation with a quadratic equation.
1. Warm-Up: Quadratic Vocabulary
Match each term with the correct description. Write the letter on the line.
- _____ Vertex
- _____ Axis of symmetry
- _____ x-intercept
- _____ y-intercept
- _____ Leading coefficient
A. The point where the graph crosses the y-axis
B. The highest or lowest point of a parabola
C. The coefficient of the highest-power term
D. A line that divides a parabola into two matching halves
E. A point where the graph crosses the x-axis
Helpful example: In (f(x)=2x^2-8x+3), the leading coefficient is 2.
2. Solve by Factoring
Solve each equation. Check each answer by substitution.
-
(x^2-9x+20=0)
Work: ____
Answer: ____
-
(x^2+2x-15=0)
Work: ____
Answer: ____
-
(2x^2-10x=0)
Work: ____
Answer: ____
-
(3x^2-12x+9=0)
Work: ____
Answer: ____
-
(x^2+6x+10=0)
Work or explanation: _____
Answer: ____
Hint: If a quadratic does not factor over the real numbers, its discriminant may be negative.
3. Choose the Best Method
For each equation, circle the method that is most efficient: factoring, square root method, completing the square, or quadratic formula. Then solve.
-
(x^2-49=0)
Method: factoring / square root method / completing the square / quadratic formula
Solution: ____
-
(x^2+8x+7=0)
Method: factoring / square root method / completing the square / quadratic formula
Solution: ____
-
(2x^2+3x-7=0)
Method: factoring / square root method / completing the square / quadratic formula
Solution: ____
-
((x-4)^2=18)
Method: factoring / square root method / completing the square / quadratic formula
Solution: ____
4. Analyze a Quadratic Function
Consider the function:
[ f(x)=x^2-6x+5 ]
-
Rewrite the function in factored form.
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Find the x-intercepts.
-
Find the y-intercept.
-
Find the axis of symmetry.
-
Find the vertex.
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Does the parabola open upward or downward? Explain how you know.
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What is the minimum or maximum value of the function?
-
Complete the table and use it to sketch the graph on the coordinate grid provided by your teacher or draw one below.
| x | f(x) |
|---|---|
| 0 | 5 |
| 1 | __ |
| 2 | __ |
| 3 | __ |
| 4 | __ |
| 5 | __ |
Sketch space:
5. Real-World Model: Garden Area
A community garden has 40 meters of fencing to enclose a rectangular planting area. Let (x) represent the width in meters.
-
Write an expression for the length in terms of (x).
Hint: The perimeter equation is (2x+2L=40).
(L=) ____
-
Write a quadratic function (A(x)) for the area of the garden.
(A(x)=) ____
-
Find the dimensions that produce the greatest possible area.
Width: ____ meters
Length: ____ meters
-
What is the greatest possible area?
____ square meters
-
Explain why the answer makes practical sense.
-
What domain makes sense for this situation? Use inequalities.
6. Projectile Motion Challenge
A ball is thrown upward from a platform. Its height is modeled by
[ h(t)=-16t^2+48t+5 ]
where (h(t)) is the height in feet and (t) is the time in seconds.
-
What is the ball’s starting height?
____ feet
-
How long does it take the ball to reach its maximum height?
____ seconds
-
What is the maximum height?
____ feet
-
Use the quadratic formula to determine when the ball reaches the ground. Round to the nearest tenth of a second.
Work: ____
Answer: ____ seconds
-
Why should one solution to the equation (h(t)=0) be rejected in this situation?
7. Error Analysis
A student solves (x^2-5x=14) as follows:
[ x(x-5)=14 ]
[ x=14 \quad \text{or} \quad x-5=14 ]
[ x=14 \quad \text{or} \quad x=19 ]
-
Identify the student’s error.
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Solve the equation correctly.
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Verify both solutions.
8. Extension: Design Your Own Model
Create a quadratic situation involving one of these contexts: profit, ticket sales, a jump, garden area, or the height of an object.
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Write a quadratic equation or function.
-
State what each variable represents.
-
Find and interpret the vertex.
-
Find any meaningful zeros or intercepts.
-
Explain one decision that could be made using your model.
Answer Key
1. Warm-Up: Quadratic Vocabulary
- B
- D
- E
- A
- C
2. Solve by Factoring
-
(x^2-9x+20=(x-4)(x-5))
Answer: (x=4) or (x=5)
-
(x^2+2x-15=(x+5)(x-3))
Answer: (x=-5) or (x=3)
-
(2x^2-10x=2x(x-5))
Answer: (x=0) or (x=5)
-
(3x^2-12x+9=3(x-1)(x-3))
Answer: (x=1) or (x=3)
-
The discriminant is (6^2-4(1)(10)=-4), which is negative.
Answer: No real solutions. In the complex number system, (x=-3\pm i).
3. Choose the Best Method
-
Square root method: (x^2=49), so (x=\pm7).
-
Factoring: ((x+1)(x+7)=0), so (x=-1) or (x=-7).
-
Quadratic formula:
[ x=\frac{-3\pm\sqrt{65}}{4} ]
-
Square root method: (x-4=\pm\sqrt{18}=\pm3\sqrt2), so (x=4\pm3\sqrt2).
4. Analyze a Quadratic Function
-
Factored form: (f(x)=(x-1)(x-5))
-
x-intercepts: ((1,0)) and ((5,0))
-
y-intercept: ((0,5))
-
Axis of symmetry: (x=3)
-
Vertex: ((3,-4))
-
The parabola opens upward because the leading coefficient is positive.
-
Minimum value: (-4)
-
Table values:
| x | f(x) |
|---|---|
| 0 | 5 |
| 1 | 0 |
| 2 | -3 |
| 3 | -4 |
| 4 | -3 |
| 5 | 0 |
5. Real-World Model: Garden Area
-
(L=20-x)
-
(A(x)=x(20-x)=-x^2+20x)
-
The vertex occurs at (x=-\frac{20}{2(-1)}=10).
Width: 10 meters
Length: 10 meters -
Greatest area: (A(10)=100) square meters
-
A square gives the greatest area when a fixed amount of fencing is used for a rectangle.
-
Meaningful domain: (0<x<20)
6. Projectile Motion Challenge
-
Starting height: 5 feet
-
Time to maximum height:
[ t=-\frac{48}{2(-16)}=1.5 ]
Answer: 1.5 seconds
-
Maximum height:
[ h(1.5)=-16(1.5)^2+48(1.5)+5=41 ]
Answer: 41 feet
-
Set the height equal to zero:
[ -16t^2+48t+5=0 ]
The solutions are approximately (t=-0.1) and (t=3.1).
Answer: The ball reaches the ground after approximately 3.1 seconds.
-
The negative solution represents a time before the ball was thrown, so it is not meaningful in this situation.
7. Error Analysis
-
The student did not set the equation equal to zero and incorrectly separated factors even though the equation was not factored as a product equal to zero.
-
Correct solution:
[ x^2-5x=14 ]
[ x^2-5x-14=0 ]
[ (x-7)(x+2)=0 ]
Answer: (x=7) or (x=-2)
-
Verification:
(7^2-5(7)=49-35=14)
((-2)^2-5(-2)=4+10=14)
8. Extension: Design Your Own Model
Answers will vary. A complete response should include a correctly written quadratic model, clearly defined variables, a correct vertex interpretation, meaningful intercepts when applicable, and a realistic decision based on the model.