TEKS A.7A Worksheets — Algebra I Graph quadratic functions on the coordinate plane

Adrian is planting bluebonnets in his Texas backyard and wants to model the height of the flowers over time. The height of the flowers can be represented by the quadratic function h(t) = -2(t - 4)(t - 2), where h is the height in centimeters and t is the number of weeks since planting. What is the x-coordinate of the vertex of the parabola, which represents the maximum height of the flowers?

1. 3 ✓
2. 4
3. 2
4. 5

Why: To find the x-coordinate of the vertex of the quadratic function h(t) = -2(t - 4)(t - 2), we first identify the roots, which are t = 2 and t = 4. The x-coordinate of the vertex can be calculated as the average of the roots: (2 + 4) / 2 = 6 / 2 = 3. Therefore, the correct answer is 3.

Anika is planning a birthday party near the Palo Duro Canyon and wants to rent a bounce house. The rental company charges a one-time fee of $50 plus $15 for each hour of rental. If Anika has a budget of $200 for the bounce house, which inequality can be used to determine the maximum number of hours h she can rent the bounce house?

1. h ≤ 10 ✓
2. h ≤ 12
3. h ≤ 13
4. h ≤ 14

Why: The cost to rent the bounce house can be expressed by the equation 50 + 15h ≤ 200. To find out how many hours Anika can rent it for, we can subtract 50 from both sides to get 15h ≤ 150. Then, divide both sides by 15, resulting in h ≤ 10. Therefore, the inequality h ≤ 10 shows the maximum number of hours she can rent the bounce house.

Aaliyah wants to design a garden in her backyard in Austin, Texas. She plans to plant a parabolic-shaped flower bed modeled by the equation y = -x^2 + 6x - 8. What is the x-coordinate of the vertex of the parabola, which represents the maximum height of the flower bed?

1. 3 ✓
2. 2
3. 4
4. 5

Why: To find the x-coordinate of the vertex of a parabola given by the equation y = ax^2 + bx + c, you can use the formula x = -b / (2a). In this case, a = -1 and b = 6. Plugging in the values, we get x = -6 / (2 * -1) = 6 / 2 = 3. Therefore, the x-coordinate of the vertex is 3.

Fatima is organizing a fundraiser at her high school in Austin, Texas, where they plan to sell cookies. The price of each cookie is x dollars, and Fatima estimates they will sell 50 cookies. If she collects a total of 100 dollars from the sales, which equation can be used to find the price of each cookie?

1. 50x = 100 ✓
2. 50x + 20 = 100
3. 50x - 20 = 100
4. 50x = 200

Why: To find the price of each cookie, we start with the total amount collected from selling cookies, which is 100 dollars. Since they plan to sell 50 cookies, we can establish the equation 50x = 100, where x is the price of each cookie. Solving for x will give us the price per cookie, confirming that the correct equation is 50x = 100.

Luis is studying the growth of bluebonnet flowers in his garden, which he modeled with the quadratic equation y = -2(x - 3)(x - 5). What are the x-intercepts of this quadratic function, which represent the points at which the flowers reach the ground level (y = 0)?

1. 3 and 5 ✓
2. 1 and 7
3. 0 and 6
4. 2 and 4

Why: To find the x-intercepts of the quadratic function y = -2(x - 3)(x - 5), set y equal to 0: 0 = -2(x - 3)(x - 5). This gives us the factorized form, showing that the x-intercepts are at x = 3 and x = 5. Therefore, the correct answer is 3 and 5.

Mika is planting a garden in her backyard in Austin, Texas. The shape of the garden is a parabolic arch described by the equation y = -2(x - 3)^2 + 8. What is the x-coordinate of the vertex of the parabola, which represents the maximum height of the arch?

1. 3 ✓
2. 4
3. 2
4. 5

Why: To find the vertex of the parabola given by the equation y = -2(x - 3)^2 + 8, we can use the vertex form of a quadratic equation, which is y = a(x - h)^2 + k. Here, h represents the x-coordinate of the vertex. From the equation, we see that h = 3. Therefore, the x-coordinate of the vertex is 3, which corresponds to the maximum height of the garden arch.

Marquise is studying the population growth of a certain Texas town over a period of years. The quadratic equation representing the population, P(t), in thousands after t years is given by P(t) = -3t^2 + 12t + 5. What is the y-intercept of the graph of this function, which represents the population when t = 0?

1. 5 ✓
2. 12
3. 0
4. -5

Why: To find the y-intercept of the quadratic function P(t) = -3t^2 + 12t + 5, we evaluate the function at t = 0. This gives us P(0) = -3(0)^2 + 12(0) + 5 = 5. Therefore, the y-intercept, which represents the population when t = 0, is 5. This means when Marquise's observation starts, the population is 5,000.

Isabella is helping her family prepare for the San Antonio Stock Show and Rodeo. They are selling a special type of barbecue sauce, and they plan to produce a quadratic model to estimate their profits based on their production costs. If the profit, P, in dollars, can be modeled by the function P(x) = -2x^2 + 16x - 24, where x represents the number of barbecue sauces produced in hundreds, which of the following values of x corresponds to the vertex of the profit function, representing the maximum profit they can achieve?

1. 4 ✓
2. 2
3. 3
4. 1

Why: The vertex of a quadratic function in the form of P(x) = ax^2 + bx + c can be found using the formula x = -b / (2a). In this case, a = -2 and b = 16. Substituting these values gives x = -16 / (2 * -2) = 4. This means that producing 400 barbecue sauces will yield the maximum profit.