The Graph Of A Quadratic Function Has A Vertex At { (4,5)$}$ And Passes Through The Point { (0,53)$}$.Write An Equation For The Function In Vertex Form And Standard Form. Show All Work.

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Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their graph is crucial for various applications in science, engineering, and economics. In this article, we will explore the graph of a quadratic function with a vertex at (4,5) and passing through the point (0,53). We will derive the equation of the function in both vertex form and standard form, providing step-by-step solutions and explanations.

Vertex Form

The vertex form of a quadratic function is given by:

f(x) = a(x - h)^2 + k

where (h, k) is the vertex of the parabola. In this case, the vertex is (4, 5), so we can write the equation as:

f(x) = a(x - 4)^2 + 5

To find the value of 'a', we can use the fact that the function passes through the point (0, 53). Substituting x = 0 and f(x) = 53 into the equation, we get:

53 = a(0 - 4)^2 + 5

Simplifying the equation, we get:

53 = 16a + 5

Subtracting 5 from both sides, we get:

48 = 16a

Dividing both sides by 16, we get:

a = 3

Now that we have found the value of 'a', we can write the equation of the function in vertex form:

f(x) = 3(x - 4)^2 + 5

Standard Form

To convert the equation from vertex form to standard form, we need to expand the squared term:

f(x) = 3(x^2 - 8x + 16) + 5

Distributing the 3, we get:

f(x) = 3x^2 - 24x + 48 + 5

Combining like terms, we get:

f(x) = 3x^2 - 24x + 53

Graph of the Function

The graph of the function f(x) = 3(x - 4)^2 + 5 is a parabola with a vertex at (4, 5). The parabola opens upward, and the axis of symmetry is the vertical line x = 4.

Properties of the Function

The function f(x) = 3(x - 4)^2 + 5 has several important properties:

  • Vertex: The vertex of the parabola is (4, 5).
  • Axis of Symmetry: The axis of symmetry is the vertical line x = 4.
  • Direction of Opening: The parabola opens upward.
  • Minimum Value: The minimum value of the function is 5, which occurs at the vertex (4, 5).
  • Maximum Value: The function does not have a maximum value, as it extends to infinity in the upward direction.

Conclusion

In this article, we have derived the equation of a quadratic function in both vertex form and standard form. We have also explored the graph of the function and its properties. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. The standard form of a quadratic function is given by f(x) = ax^2 + bx + c, where a, b, and c are constants. Understanding the graph of a quadratic function is crucial for various applications in science, engineering, and economics.

References

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Frequently Asked Questions

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Q: How do I find the value of 'a' in the vertex form of a quadratic function?

A: To find the value of 'a', you can use the fact that the function passes through a given point. Substitute the x and y values of the point into the equation and solve for 'a'.

Q: What is the standard form of a quadratic function?

A: The standard form of a quadratic function is given by f(x) = ax^2 + bx + c, where a, b, and c are constants.

Q: How do I convert the vertex form of a quadratic function to standard form?

A: To convert the vertex form to standard form, expand the squared term and distribute the coefficient.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is the vertical line that passes through the vertex of the parabola.

Q: Does a quadratic function have a maximum value?

A: No, a quadratic function does not have a maximum value, as it extends to infinity in the upward direction.

Q: Can a quadratic function have a minimum value?

A: Yes, a quadratic function can have a minimum value, which occurs at the vertex of the parabola.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, use the formula (h, k) = (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the standard form of the function.

Q: What is the relationship between the vertex form and standard form of a quadratic function?

A: The vertex form and standard form of a quadratic function are equivalent, but they represent the function in different ways.

Q: Can a quadratic function be represented in other forms?

A: Yes, a quadratic function can be represented in other forms, such as factored form or polynomial form.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, use the vertex form and plot the vertex, then use the axis of symmetry to find the other points on the graph.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, such as modeling the trajectory of a projectile, the motion of an object under constant acceleration, and the growth of a population.

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Practice Problems

  1. Find the vertex form of the quadratic function f(x) = 2x^2 + 12x + 10.
  2. Convert the vertex form of the quadratic function f(x) = (x - 2)^2 + 3 to standard form.
  3. Find the axis of symmetry of the quadratic function f(x) = x^2 + 4x + 4.
  4. Determine whether the quadratic function f(x) = x^2 + 2x + 1 has a maximum value or a minimum value.
  5. Find the vertex of the quadratic function f(x) = x^2 + 6x + 8.

Answers

  1. f(x) = 2(x + 3)^2 + 1
  2. f(x) = 2x^2 + 4x + 1
  3. x = -2
  4. The quadratic function has a minimum value.
  5. (h, k) = (-3, 1)