Problem 40 - 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: (4,1) Point: (6,-7)

Step-by-Step Solution

Answer

The standard form of the quadratic function given that the vertex is (4,1) and the graph passes through the point (6,-7) is \(f(x) = -2(x-4)^2 + 1\).

Step 1: Start With The General Form

Begin with the standard form of a quadratic function, which is \(f(x) = a(x-h)^2 + k \). Here, h and k refer to the x and y coordinates of the vertex respectively. These are provided as (4,1), so substitute those values into the equation to get \(f(x) = a(x-4)^2 + 1 \). However, the equation is not complete as a is still undetermined.

Step 2: Substituting The Point Given Into The Equation

To find the value of a, use the point that the graph passes through (6,-7). Substitute x = 6 and f(x) = -7 into the equation and then solve for a. This gives: \(-7 = a(6-4)^2 + 1\). Subtract 1 from both sides, we have \(-8 = a(2)^2\), which simplifies to \(-8 = 4a\).

Step 3: Solve for a

The next step here is to solve for a by dividing both sides of the equation by 4: \(-8/4 = a\). This simplifies to \(a = -2\).

Step 4: Compute The Final Equation

Now that we have the value of a as -2, insert this into our initial standard form equation. This completes the quadratic function, which is: \(f(x) = -2(x-4)^2 + 1\). This is the standard form of the given quadratic function.

Key Concepts

Vertex Form of Quadratic Function

The vertex form of a quadratic function makes it easy to identify key features of the quadratic graph, specifically the vertex, which is the highest or lowest point. The formula for a quadratic in vertex form is given by \[ f(x) = a(x-h)^2 + k \] where

• \( a \) is a coefficient that affects the direction and width of the parabola.
• \( (h, k) \) represents the vertex of the parabola.

By using the vertex form, you can directly read the vertex coordinates \( (h, k) \) from the equation. This is useful for easily sketching the graph of the quadratic without any transformations.

Solving for Parameters

To fully define a quadratic function in vertex form, you need to solve for any unknown parameters. Given a vertex and a point that the function passes through, you can find the coefficient \( a \). Here's a step-by-step process:1. **Start with the vertex form equation**: Substitute the vertex coordinates into the equation: \( f(x) = a(x-h)^2 + k \).
2. **Use the given point**: Substitute the x and y values from the given point into the equation: \( y = a(x-h)^2 + k \).
3. **Solve for \( a \)**: Expand and solve the resulting equation to find the value of \( a \).
This step is essential because the value of \( a \) determines the shape and orientation of the parabola.

Graphing Quadratic Equations

Graphing a quadratic function involves plotting its vertex and symmetry, and determining its intercepts. When the quadratic is in vertex form, the process simplifies significantly:

• **Locate the Vertex**: At the point \( (h, k) \).
• **Identify the Axis of Symmetry**: This is a vertical line that passes through the vertex, given by \( x = h \).
• **Plot Additional Points**: Use symmetry and another given point or various x-values to figure out points around the vertex.
• **Sketch the Parabola**: Using \( a \), determine if the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).

Put all information together to sketch the graph accurately, ensuring it passes through all calculated points and reflects the calculated characteristics.

Vertex of a Quadratic

The vertex of a quadratic function is a pivotal point that determines the function's appearance on a graph. As the point \( (h, k) \), the vertex represents:

• **Peak or Trough**: If \( a > 0 \), the vertex is a minimum point. If \( a < 0 \), it's a maximum point.
• **Axis of Symmetry**: Passes through this point, ensuring that the parabola is symmetric around this vertical line.

Understanding the vertex is crucial when analyzing the behavior of a quadratic graph. It gives insights into the parabola's position without requiring extensive calculations and highlights the quadratic function's key characteristics.