Graphing Quadratic Functions and Key Features

1) What is the vertex of the function f(x) = x^2 + 2x - 3?

(-1, -4)

(1, -2)

(0, -3)

(2, 1)

To find the vertex of the quadratic function f(x) = x^2 + 2x - 3, we can use the vertex formula, which states that the x-coordinate of the vertex is given by x = -b/(2a). Here, a = 1 and b = 2, so x = -2/(2*1) = -1. Substituting x = -1 back into the function gives f(-1) = (-1)^2 + 2(-1) - 3 = 1 - 2 - 3 = -4. Therefore, the vertex of the function is (-1, -4).

Explanation

To find the vertex of the quadratic function f(x) = x^2 + 2x - 3, we can use the vertex formula, which states that the x-coordinate of the vertex is given by x = -b/(2a). Here, a = 1 and b = 2, so x = -2/(2*1) = -1. Substituting x = -1 back into the function gives f(-1) = (-1)^2 + 2(-1) - 3 = 1 - 2 - 3 = -4. Therefore, the vertex of the function is (-1, -4).

2) What is the axis of symmetry for the function f(x) = -x^2 - 10x - 25?

-5

5

10

-10

The axis of symmetry for a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b/(2a). In this case, a = -1 and b = -10. Plugging in these values gives x = -(-10)/(2 * -1) = 10/-2 = -5. This means the graph of the function is symmetric about the line x = -5, which is the axis of symmetry.

Explanation

The axis of symmetry for a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b/(2a). In this case, a = -1 and b = -10. Plugging in these values gives x = -(-10)/(2 * -1) = 10/-2 = -5. This means the graph of the function is symmetric about the line x = -5, which is the axis of symmetry.

3) What is the y-intercept of the function f(x) = x^2 - 6x + 8?

8

0

-8

6

To find the y-intercept of the function f(x) = x^2 - 6x + 8, we evaluate the function at x = 0. Substituting 0 into the equation gives f(0) = 0^2 - 6(0) + 8, which simplifies to f(0) = 8. Therefore, the y-intercept, where the graph crosses the y-axis, is 8.

Explanation

To find the y-intercept of the function f(x) = x^2 - 6x + 8, we evaluate the function at x = 0. Substituting 0 into the equation gives f(0) = 0^2 - 6(0) + 8, which simplifies to f(0) = 8. Therefore, the y-intercept, where the graph crosses the y-axis, is 8.

4) Which of the following functions opens downwards?

F(x) = x^2 + 2x - 3

F(x) = -x^2 - 10x - 25

F(x) = x^2 - 6x + 8

F(x) = x^2

A function opens downwards if its leading coefficient is negative. In the given options, the function f(x) = -x^2 - 10x - 25 has a leading coefficient of -1, which is negative. This indicates that the parabola formed by this quadratic function will open downwards, resulting in a maximum point. In contrast, the other functions have positive leading coefficients, causing them to open upwards. Thus, f(x) = -x^2 - 10x - 25 is the only function that opens downwards.

Explanation

A function opens downwards if its leading coefficient is negative. In the given options, the function f(x) = -x^2 - 10x - 25 has a leading coefficient of -1, which is negative. This indicates that the parabola formed by this quadratic function will open downwards, resulting in a maximum point. In contrast, the other functions have positive leading coefficients, causing them to open upwards. Thus, f(x) = -x^2 - 10x - 25 is the only function that opens downwards.

5) What is the minimum value of the function f(x) = x^2 + 2x - 3?

-4

0

1

-3

To find the minimum value of the quadratic function f(x) = x^2 + 2x - 3, we can complete the square or use the vertex formula. The function is a parabola opening upwards, and its vertex represents the minimum point. By completing the square, we rewrite the function as f(x) = (x + 1)^2 - 4. The vertex occurs at x = -1, where f(-1) = -4. Thus, the minimum value of the function is -4.

Explanation

To find the minimum value of the quadratic function f(x) = x^2 + 2x - 3, we can complete the square or use the vertex formula. The function is a parabola opening upwards, and its vertex represents the minimum point. By completing the square, we rewrite the function as f(x) = (x + 1)^2 - 4. The vertex occurs at x = -1, where f(-1) = -4. Thus, the minimum value of the function is -4.

6) What are the roots of the function f(x) = x^2 - 6x + 8?

2 and 4

0 and 8

1 and 7

3 and 3

To find the roots of the quadratic function f(x) = x^2 - 6x + 8, we can factor the expression. It can be rewritten as (x - 2)(x - 4) = 0. Setting each factor equal to zero gives us the solutions x - 2 = 0 and x - 4 = 0, leading to the roots x = 2 and x = 4. These values indicate where the function intersects the x-axis.

Explanation

To find the roots of the quadratic function f(x) = x^2 - 6x + 8, we can factor the expression. It can be rewritten as (x - 2)(x - 4) = 0. Setting each factor equal to zero gives us the solutions x - 2 = 0 and x - 4 = 0, leading to the roots x = 2 and x = 4. These values indicate where the function intersects the x-axis.

7) What is the leading coefficient of the function f(x) = -x^2 - 10x - 25?

-1

1

10

-10

The leading coefficient of a polynomial function is the coefficient of the term with the highest power of x. In the function f(x) = -x^2 - 10x - 25, the term with the highest power is -x^2. The coefficient of this term is -1, which indicates that the graph of the function opens downwards. Thus, the leading coefficient is -1.

Explanation

The leading coefficient of a polynomial function is the coefficient of the term with the highest power of x. In the function f(x) = -x^2 - 10x - 25, the term with the highest power is -x^2. The coefficient of this term is -1, which indicates that the graph of the function opens downwards. Thus, the leading coefficient is -1.

8) What is the vertex of the function f(x) = x^2 - 6x + 8?

(3, -1)

(3, 1)

(0, 8)

(6, 0)

To find the vertex of the quadratic function f(x) = x^2 - 6x + 8, we can use the vertex formula, which is given by the coordinates (h, k) where h = -b/(2a). Here, a = 1 and b = -6, so h = 6/2 = 3. To find k, we substitute x = 3 back into the function: f(3) = 3^2 - 6(3) + 8 = 9 - 18 + 8 = -1. Thus, the vertex is at the point (3, -1).

Explanation

To find the vertex of the quadratic function f(x) = x^2 - 6x + 8, we can use the vertex formula, which is given by the coordinates (h, k) where h = -b/(2a). Here, a = 1 and b = -6, so h = 6/2 = 3. To find k, we substitute x = 3 back into the function: f(3) = 3^2 - 6(3) + 8 = 9 - 18 + 8 = -1. Thus, the vertex is at the point (3, -1).