Problem 39 - Algebra and Trigonometry

Question

Find a function whose graph is a parabola with vertex \((1,-2)\) and that passes through the point \((4,16) .\)

Step-by-Step Solution

Answer

The function is \( y = 2(x-1)^2 - 2 \).

Step 1: Identify the Parabola's Vertex Form

A parabola can be expressed in vertex form as \( y = a(x-h)^2 + k \). Here, \((h, k)\) is the vertex of the parabola. Since the given vertex is \((1, -2)\), we can substitute \(h = 1\) and \(k = -2\) into the equation.

Step 2: Write the Initial Equation

Substituting the vertex \((1, -2)\) into the vertex form equation, we have \( y = a(x-1)^2 - 2 \). The only unknown here is \(a\), which we need to find by using the point \((4, 16)\) that lies on the parabola.

Step 3: Substitute the Given Point into the Equation

Since the parabola passes through the point \((4, 16)\), we let \(x = 4\) and \(y = 16\) in the equation \( y = a(x-1)^2 - 2 \). This will allow us to solve for \(a\).

Step 4: Solve for the Parameter a

Substituting these values gives us: \( 16 = a(4-1)^2 - 2 \). This simplifies to \( 16 = a(3)^2 - 2 \) or \( 16 = 9a - 2 \). Adding 2 to both sides yields \( 18 = 9a \). Dividing by 9 gives \( a = 2 \).

Step 5: Write the Complete Equation of the Parabola

Now that we have found \(a = 2\), substitute it back into the vertex form equation: \( y = 2(x-1)^2 - 2 \). This is the equation of the parabola with the specified characteristics.

Key Concepts

Vertex

The vertex of a parabola is a crucial point which either represents its highest or lowest point. It essentially marks the turning point of the parabola's curve. Think of it as the tip of a "U" shape when a parabola opens upwards or downwards. Understanding the vertex helps in easily identifying the orientation and position of the parabola.

• The vertex is represented by the coordinates \( (h, k) \), where \( h \) is the x-coordinate and \( k \) is the y-coordinate.
• In the context of the problem, the vertex given is \( (1, -2) \).
• Knowing the vertex helps in writing the equation of the parabola directly in vertex form.

Parabola Equation

A parabola is represented by a quadratic equation and can take different forms depending on the information given. One such form is the vertex form, which makes use of the vertex coordinates to express the equation.

• Each parabola follows the path of the quadratic equation \( ax^2 + bx + c = y \) in standard form.
• The vertex form, however, simplifies this to \( y = a(x-h)^2 + k \), allowing us to use the \( (h, k) \) vertex easily.
• This format provides clarity, especially when transformations are being considered such as shifts or stretches.

By substituting values into this form, the behavior of the parabola becomes more intuitive and manageable.

Vertex Form of a Parabola

The vertex form, \( y = a(x-h)^2 + k \), provides a clear representation of a parabola by using its vertex. This form is incredibly useful in solving various mathematical problems concerning parabolas.

• This format easily indicates the transformations applied to the graph of the parabola, such as vertex shifting.
• In this problem, the vertex form becomes \( y = a(x-1)^2 - 2 \) after substituting the vertex \( (1, -2) \).
• The value of \( a \) in this formula affects the parabola's width and direction.

With this form, predicting how the parabola behaves, such as identifying its direction and size, becomes a straightforward task.

Solving for Parameter a

To completely define the parabola equation in vertex form, the value of \( a \) needs to be calculated. This parameter dictates how "steep" or "wide" the parabola is, and whether it opens upwards or downwards.

• For our problem, knowing the point \( (4, 16) \) on the parabola provides necessary information to determine \( a \).
• Substitute \( x = 4 \) and \( y = 16 \) into \( y = a(x-1)^2 - 2 \), simplifying to solve for \( a \).
• This results in \( 16 = 9a - 2 \), which simplifies to \( a = 2 \) after some algebraic manipulation.

By finding the value of \( a \), we complete the parabola's equation: \( y = 2(x-1)^2 - 2 \). This describes how tightly the parabola curves, as well as its overall orientation.