TEKS A.6B Worksheets — Algebra I Write equations of quadratic functions given the

Ximena is planning a garden in her backyard in Houston, Texas, and wants to model the shape of a parabolic flower bed. The vertex of the flower bed is located at the point (2, 3), and another point on the edge of the flower bed is at (4, 7). What is the equation of the flower bed in vertex form?

1. f(x) = 2(x - 2)² + 3
2. f(x) = 1/2(x - 2)² + 3
3. f(x) = (x - 2)² + 3 ✓
4. f(x) = 3(x - 2)² + 3

Why: To find the equation in vertex form, we start with the vertex (h, k) = (2, 3). We can use the point (4, 7) to solve for 'a'. The vertex form is f(x) = a(x - h)² + k. Plugging in (4, 7), we have 7 = a(4 - 2)² + 3. This simplifies to 7 = a(2)² + 3, or 7 = 4a + 3. Solving for 'a', we get 4 = 4a, so a = 1. Therefore, the equation in vertex form is f(x) = 1(x - 2)² + 3, which can be written simply as f(x) = (x - 2)² + 3.

Elijah is designing a garden in his backyard in Fort Worth, Texas, using a quadratic function to represent the height of a plant over time. The vertex of the graph is at the point (2, 3), meaning the optimal growth occurs at 2 weeks with a height of 3 inches. After 4 weeks, the plant reaches a height of 7 inches. Using the vertex form of the quadratic function, what is the equation of the plant's height as a function of time in standard form?

1. x² - 4x + 7 ✓
2. x² + 4x - 1
3. x² - 4x + 3
4. x² + 4x + 1

Why: To find the equation of the quadratic function, we need to determine the value of 'a' in the vertex form of the equation f(x) = a(x - h)² + k, where (h, k) is the vertex. The vertex here is (2, 3), so the function starts as f(x) = a(x - 2)² + 3. We also know that f(4) = 7, since the height after 4 weeks is 7 inches. Plugging in those values into the equation gives us 7 = a(4 - 2)² + 3. Simplifying that, we have 7 = 4a + 3, so 4 = 4a, which means a = 1. Thus, the vertex form becomes f(x) = (x - 2)² + 3. Expanding this to standard form, we get f(x) = (x² - 4x + 4) + 3, which simplifies to f(x) = x² - 4x + 7.

Karim is building a small garden in his backyard in Nacogdoches, Texas. He wants the garden to have a parabolic shape with its vertex at the point (3, 2). If another point on the parabola is (4, 3), what is the equation of the parabola in vertex form, f(x) = a(x - h)² + k?

1. f(x) = 1(x - 3)² + 2 ✓
2. f(x) = 2(x - 3)² + 2
3. f(x) = 1/2(x - 3)² + 2
4. f(x) = 3(x - 3)² + 2

Why: To find the equation in vertex form, we start with the vertex (h, k) = (3, 2). The form is f(x) = a(x - 3)² + 2. We can find 'a' by substituting the other point (4, 3) into the equation. Replacing f(x) with 3, we have 3 = a(4 - 3)² + 2. This simplifies to 3 = a(1) + 2, leading to a = 3 - 2, which means a = 1. Therefore, the equation is f(x) = 1(x - 3)² + 2.

In a Texas ranch, Madison is trying to design a fence for a rectangular area where she will keep her cattle. The length of the fence is represented by the expression 2x + 4, and the width is represented by the expression x + 2. If the area of the rectangular space must be 60 square meters, which equation can be used to determine the value of x?

1. (2x + 4)(x + 2) = 60 ✓
2. (2x + 4) + (x + 2) = 60
3. (2x + 4) - (x + 2) = 60
4. (2x + 4)(x + 2) = 120

Why: To find the area of the rectangle, we use the formula for area, which is length times width. The length is given by 2x + 4 and the width by x + 2. Therefore, the equation representing the area can be expressed as (2x + 4)(x + 2) = 60. This means the correct choice is the first one, which sets up the equation correctly to solve for x.

Carolina is designing a park near Enchanted Rock and wants to create a parabolic path for visitors to enjoy the scenery. The vertex of the path is at (2, 3), and it passes through the point (4, 7). What is the equation of the path in standard form?

1. x² - 4x + 7 ✓
2. x² - 2x + 5
3. 2x² - 8x + 3
4. 3x² - 6x + 1

Why: To find the equation in standard form, start with the vertex form f(x) = a(x - h)² + k, where (h, k) is the vertex. Here, h = 2 and k = 3, giving f(x) = a(x - 2)² + 3. To find 'a', use the point (4, 7): 7 = a(4 - 2)² + 3. This simplifies to 7 = 4a + 3. Solving for 'a' gives 4a = 4, so a = 1. Therefore, the equation is f(x) = (x - 2)² + 3. Expanding this gives f(x) = x² - 4x + 4 + 3 = x² - 4x + 7. Thus, the equation in standard form is x² - 4x + 7.

Caroline is organizing a fundraiser at her school in the DFW Metroplex to support a local wildlife conservation project. The amount of money she raises can be modeled by the quadratic equation f(x) = -2(x - 3)² + 24, where x represents the number of hours spent fundraising. What is the maximum amount of money Caroline can raise during her fundraiser?

1. 24 ✓
2. 12
3. 20
4. 16

Why: To find the maximum amount of money Caroline can raise, we can evaluate the vertex form of the equation, which is f(x) = -2(x - 3)² + 24. The vertex of this quadratic function gives us the maximum value of f(x) when x = 3. Plugging in x = 3, we find f(3) = -2(3 - 3)² + 24 = -2(0) + 24 = 24. Thus, the maximum amount of money Caroline can raise is 24.

Janelle is designing a decorative archway for the San Antonio Stock Show. The vertex of the archway is at the point (2, 3), and it passes through the point (4, 7). What is the equation of the archway in vertex form?

1. f(x) = 2(x - 2)² + 3
2. f(x) = 1(x - 2)² + 3 ✓
3. f(x) = 4(x - 2)² + 3
4. f(x) = 3(x - 2)² + 3

Why: To find the equation of a quadratic function in vertex form, we use the vertex (h, k) = (2, 3) and plug it into the equation f(x) = a(x - h)² + k. We know the vertex is (2, 3), so we have f(x) = a(x - 2)² + 3. Next, we need to determine 'a' using the point (4, 7). Substituting x = 4 and f(4) = 7 gives us: 7 = a(4 - 2)² + 3. This simplifies to 7 = a(2)² + 3, or 7 = 4a + 3. By solving for 'a', we have 4 = 4a, thus a = 1. Therefore, the equation of Janelle's archway is f(x) = 1(x - 2)² + 3.

Andrea is organizing a bake sale at her school in Texas to raise funds for a local charity. She knows that the number of cookies she bakes can be modeled by the quadratic equation f(x) = 2(x - 3)² + 5, where x represents the number of hours she spends baking. What is the standard form of this quadratic function?

1. 2x² - 12x + 23 ✓
2. 2x² + 12x + 23
3. 2x² - 12x - 23
4. 2x² + 12x - 23

Why: To convert the vertex form f(x) = 2(x - 3)² + 5 to standard form, first expand the squared term: (x - 3)² = x² - 6x + 9. Then apply the 2 to get 2(x² - 6x + 9) = 2x² - 12x + 18. Finally, add 5 to get f(x) = 2x² - 12x + 23. Therefore, the correct answer is 2x² - 12x + 23.