TEKS A.6C Worksheets — Algebra I Write quadratic functions when given real solutions

Madison is planning to grow a peach orchard in the Rio Grande Valley. She believes that the height of the peach trees, in feet, can be modeled by the quadratic equation h(t) = (t - 2)(t - 8), where t represents the number of years since planting. Which of the following expressions represents the height of the trees as a function of time and can be used to find the height when t = 4?

1. t^2 - 10t + 16 ✓
2. t^2 - 6t + 16
3. t^2 - 10t + 8
4. t^2 - 6t + 8

Why: To find the quadratic expression from the factored form (t - 2)(t - 8), we need to expand it. Distributing, we have: t * t - 8t - 2t + 16, which simplifies to t^2 - 10t + 16. Therefore, the expression that represents the height of the trees is t^2 - 10t + 16, making the correct answer the first choice.

Antonio is helping his family on their farm in Central Texas, where they grow tomatoes. Last season, they planted a total of 60 tomato plants. This year, they plan to plant x fewer tomato plants than last season, and they want the number of plants this year to be represented by the equation y = (x - 10)(x + 6). If they want to know how many plants they are planning to plant this year, which expression can be used to determine the number of plants based on the value of x?

1. x^2 - 4x - 60 ✓
2. x^2 + 4x - 60
3. x^2 - 4x + 60
4. x^2 + 4x + 60

Why: To find the number of plants represented by the equation y = (x - 10)(x + 6), we need to expand this expression. When we use the distributive property (FOIL method), we get: y = x^2 + 6x - 10x - 60, which simplifies to y = x^2 - 4x - 60. Therefore, the correct choice is the first option.

Emilio is planting bluebonnets in his garden located in the Texas Hill Country. He discovers that the number of bluebonnet plants he can fit in a rectangular garden is represented by the quadratic function f(x) = x^2 - 10x + 24, where x is the number of plants per row. If Emilio wants to determine the number of plants he can fit in each row, which equation can he use to find the solutions for the number of rows he can plant in the garden (in factored form)?

1. (x - 6)(x - 4) ✓
2. (x - 12)(x + 2)
3. (x - 8)(x - 3)
4. (x + 6)(x - 4)

Why: To find the number of plants he can fit in each row, we need to factor the quadratic equation f(x) = x^2 - 10x + 24. By finding two numbers that multiply to 24 and add up to -10, we find that these numbers are -6 and -4. Thus, the factored form is (x - 6)(x - 4). This means Emilio can fit 6 or 4 bluebonnets in each row.

Gabriela is creating a new community garden in Austin, Texas. The height of the plants in her garden can be modeled by the quadratic function h(x) = x^2 - 7x + 10, where x represents the number of weeks since planting. What is the factored form of the height equation, h(x), based on its real solutions?

1. (x - 2)(x - 5) ✓
2. (x + 2)(x + 5)
3. (x - 1)(x - 10)
4. (x + 1)(x - 10)

Why: To find the factored form of the quadratic function h(x) = x^2 - 7x + 10, we first look for the roots by solving the equation x^2 - 7x + 10 = 0. Factoring gives us (x - 2)(x - 5) = 0, indicating the roots are x = 2 and x = 5. Therefore, the factored form of the height equation is (x - 2)(x - 5).

Carolina is designing a rectangular garden in her backyard in Austin, Texas. The length of the garden is 3 feet longer than twice the width. If the width of the garden is x feet, which quadratic function represents the area of the garden in square feet when expressed in terms of x?

1. 2x^2 + 3x ✓
2. 2x^2 + 3x - 6
3. 2x^2 + 3x + 6
4. x^2 + 3x

Why: To find the area of the rectangular garden, we use the formula for area, A = length * width. Given that the length is 3 feet longer than twice the width, we can express the length as 2x + 3. Therefore, the area A can be written as A = (2x + 3) * x = 2x^2 + 3x. The correct quadratic function that represents the area of the garden in square feet is 2x^2 + 3x.

Lucy is studying the growth of a sunflower garden in New Braunfels, Texas. After several weeks, she finds that the height of her sunflowers can be modeled by the equation h(x) = (x - 3)(x - 5), where h(x) is the height in centimeters and x is the number of weeks since she planted them. What is the quadratic function in standard form that represents the height of the sunflowers over time?

1. x^2 - 8x + 15 ✓
2. x^2 + 8x - 15
3. x^2 - 15
4. x^2 + 15

Why: To find the quadratic function in standard form, we start with the factored form h(x) = (x - 3)(x - 5). We can expand this by using the distributive property: h(x) = x^2 - 5x - 3x + 15, which simplifies to h(x) = x^2 - 8x + 15. Therefore, the quadratic function in standard form is x^2 - 8x + 15.

Emilio is studying the growth of bluebonnet flowers in his Texan backyard. He observes that the height of the flowers can be modeled by a quadratic function. He finds that one type of bluebonnet reaches a height of 0 inches at 2 inches and 5 inches. Which quadratic function in factored form best represents the height of the bluebonnet flowers based on these heights?

1. h(x) = (x - 2)(x - 5) ✓
2. h(x) = (x + 2)(x + 5)
3. h(x) = (x - 2)(x + 5)
4. h(x) = (x + 2)(x - 5)

Why: The correct answer is h(x) = (x - 2)(x - 5). In factored form, a quadratic function is expressed as f(x) = (x - r1)(x - r2), where r1 and r2 are the roots or x-intercepts. Since the bluebonnet flowers have heights of 0 at 2 inches and 5 inches, the roots are x = 2 and x = 5. Therefore, the quadratic function that represents this situation in factored form is (x - 2)(x - 5).

Valentina is studying the growth of prickly pear cacti in Caddo Lake, Texas. She observes that the number of cacti can be modeled by the equation y = (x - 2)(x - 5), where x represents the number of years since she planted them, and y represents the total number of cacti. Which of the following represents the correct quadratic function in standard form for the number of cacti over time?

1. x^2 - 7x + 10 ✓
2. x^2 + 7x - 10
3. x^2 - 3x + 10
4. x^2 + 3x - 10

Why: To convert the factored form y = (x - 2)(x - 5) into standard form, we can apply the distributive property. First, we multiply x by both terms: x*x + x*(-5) + (-2)*x + (-2)*(-5), which results in x^2 - 5x - 2x + 10. Combining like terms gives us x^2 - 7x + 10. Therefore, the correct quadratic function in standard form is x^2 - 7x + 10.