Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result. Vertex: (-2,5) Point: (0,9)

Step-by-Step Solution

Answer

The standard form of the quadratic function that passes through the point (0,9) and has a vertex of (-2,5) is \(f(x) = x^2 + 4x + 9\).

Step 1: Find the 'a' Value

Start by substituting the given point according to the vertex form of a quadratic function. The coordinate corresponding to x is 0 and for y is 9. The vertex are -2 for 'h' and 5 for 'k'. The equation is: \(9 = a(0-(-2))^2 + 5\). Now, solve for 'a'. It yields 'a' = 1 after simplification.

Step 2: Convert to Standard Form

The equation is, applied to the vertex form, now \(f(x) = (x-(-2))^2 + 5\). This expands to \(f(x) = (x+2)^2 + 5\). Expanding the quadratic, we get \(f(x) = x^2 + 4x + 4 + 5\). Simplifying this gives the final answer: \(f(x) = x^2 + 4x + 9\). This is the standard form of a quadratic function.

Step 3: Verification

The graphing utility can be used to verify the result. As this is a text-based solution, the verification process isn't included in this step by step guideline. It involves inputting the standard equation, \(f(x) = x^2 + 4x + 9\), on the graphing calculator and checking if the graph goes through the point (0, 9) and has (-2, 5) as the vertex.

Key Concepts

Vertex Form

The vertex form of a quadratic function is one of the most helpful representations you can use. It allows you to easily identify the vertex, or the highest or lowest point, of the parabola (a u-shaped curve). The vertex form is expressed as:\[ f(x) = a(x - h)^2 + k \]- Here, \( \text{(h, k)} \) is the vertex of the parabola.- The variable \( a \) determines the direction (upward or downward) and the width of the parabola. - If \( a > 0 \), the parabola opens upwards. - If \( a < 0 \), it opens downwards.Knowing the vertex form makes it simple to sketch the basic shape and orientation of the quadratic function without needing to do much calculation. For instance, if given the vertex \((-2, 5)\), and a point \((0, 9)\), you can use these to determine the specific value of \( a \), and construct the equation into vertex form. Once you have these values, shifting to the standard form becomes straightforward.

Standard Form

The standard form of a quadratic equation is another widely used representation. It provides a linear combination of constants and coefficients, making it easy to perform algebraic operations. The standard form is given as:\[ f(x) = ax^2 + bx + c \]- Here, \( a \), \( b \), and \( c \) are constants.- The value of \( a \) is the same as in vertex form and affects the width and direction of the parabola.- \( b \) influences the position of the vertex and the symmetry of the graph, while \( c \) represents the y-intercept of the parabola.Converting from vertex form to standard form involves expanding the equation by carrying out operations such as squaring the binomial and combining like terms. This isn't just a numerical exercise but provides insights into the quadratic's graphical representation and facilitates computation, particularly when verifying your results using graphing utilities.

Graphing Utility

A graphing utility is an essential tool for visualizing and verifying quadratic functions. Graphing utilities, such as online graphing calculators or calculator software, can plot the equation, allowing you to see the parabola's shape, size, and position on a coordinate plane. - It helps in confirming the accuracy of your algebraic solution. - By inputting the quadratic function into the utility, you can check if the resulting graph: - Passes through given coordinates, such as a specific point. - Aligns perfectly with calculated vertex points. Using a graphing utility is straightforward. Once the function is input, observe how the parabola behaves. This not only confirms your solution but also enhances comprehension by allowing you to interact with the quadratic function visually.