Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex: \((-2,-2) ;\) point: \((-1,0)\)

Step-by-Step Solution

Answer

The equation of the parabola is \(y=2(x+2)^2 -2\)

Step 1: Substitute Vertex into the Equation

Substitute the vertex coordinates, \(-2\) for \(h\) and \(-2\) for \(k\) in the equation. We get \(y=a(x+2)^2-2\)

Step 2: Substitute Given Point into the Equation

Substitute the given point coordinates to solve for \(a\). We have \((x,y)=(-1,0)\). Substituting these into the equation we get \(0=a(-1+2)^2 - 2\).

Step 3: Solve for a

Solving the equation \(0=a(-1+2)^2 - 2\) for \(a\) gives \(a=2\)

Step 4: Write the Parabola Equation

Substitute the computed value of \(a\) into the equation from step 1. The equation of the parabola is \(y=2(x+2)^2 - 2\)

Key Concepts

Vertex Form

The vertex form of a parabola's equation is incredibly useful when dealing with quadratic functions. It is typically expressed as:

• \(y = a(x - h)^2 + k\)

This form provides immediate insights into the parabola's geometry:

• \(a\) influences the direction and width of the parabola. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards. The absolute value of \(a\) affects how "wide" or "narrow" the parabola will be.
• \((h, k)\) represents the coordinates of the vertex, the peak or trough of the parabola, depending on its orientation.

Switching a standard quadratic equation into vertex form can help easily identify these key features. Vertex form helps visualize the quadratic function, providing learners with an intuitive grasp of the function's behavior.

Vertex Coordinates

Vertex coordinates are an essential part of understanding quadratic functions. These coordinates tell us the exact point where the parabola changes direction, known as the vertex. In the general parabolic equation in vertex form, \(y = a(x - h)^2 + k\), the vertex is denoted by \((h, k)\).

The vertex coordinates define crucial attributes of the parabola:

• Turning Point: The x-value of the vertex, \(h\), is the line of symmetry of the parabola. Left and right from this point, the parabola's arms are "mirrors" of each other.
• Maximum or Minimum: Depending on the sign of \(a\), the vertex indicates the parabola's highest (if \(a < 0\)) or lowest point (if \(a > 0\)).

For example, the vertex \((-2, -2)\) in our specific problem tells us the parabola reaches its lowest point at this coordinate, indicating a minimum since \(a\) is positive. Understanding vertex coordinates is crucial for correctly interpreting the curve's overall direction and positioning.

Quadratic Functions

Quadratic functions are a fundamental element of algebra, characterized by their distinguishing parabolic graphs. The general form of a quadratic function is \(y = ax^2 + bx + c\). However, the vertex form \(y = a(x - h)^2 + k\) is often more useful for identifying specific properties quickly and easily.

Quadratic functions can:

• Produce U-shaped graphs called parabolas which can open upwards or downwards depending on the sign of \(a\).
• Have countless applications across fields, including physics, engineering, and finance, due to their ability to model various phenomena like projectile motion or profit maximization.
• Automatically offer insight into crucial characteristics like the vertex, line of symmetry, and direction of opening when written in vertex form.

Through their vertex and intercepts, quadratic functions provide a rich ground for analysis and problem-solving. They allow math enthusiasts to delve into the nature of curves and explore deep mathematical relationships that appear in myriad real-world contexts.