Problem 45 - Precalculus: Functions and Graphs

Question

Find an equation for the indicated half of the parabola. Upper half of \((y-5)^{2}=x+2\)

Step-by-Step Solution

Answer

The equation for the upper half is \(y = \sqrt{x + 2} + 5\).

Step 1: Identify the given equation of the parabola

The given equation is \[(y - 5)^2 = x + 2.\]This is a standard form for a parabola that opens horizontally. In this case, it can open either to the left or right.

Step 2: Determine the orientation of the parabola

Because the equation is in the form \[(y - k)^2 = a(x - h),\] where in this case, \(h = -2\) and \(k = 5,\) this parabola opens horizontally either to the left or the right depending on the value of \(a.\)In the given equation, \(a = 1,\) indicating that the parabola opens to the right.

Step 3: Rewrite equation for the whole parabola

To express the general equation of the parabola purely in terms of \(y,\) rewrite \[x = (y - 5)^2 - 2.\]

Step 4: Find the equation for the upper half

Since the parabola opens horizontally, the "upper" half means the top part of the parabola when viewed horizontally. For this part, \(y \) would be greater than \(5.\) Thus, we solve for \(y:\)\[y - 5 = \sqrt{x + 2},\]leading to the equation:\[y = \sqrt{x + 2} + 5.\]This represents the upper half where \(y \geq 5\).

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental concept in algebra and mathematics. They involve equations of the form \(ax^2 + bx + c = 0\). What makes them unique is that they graph as parabolas. These parabolas can open upwards or downwards when working with vertical quadratics. The position and orientation of a parabola are directly influenced by the coefficients \(a\), \(b\), and \(c\).
When dealing with quadratic functions, it's important to grasp concepts like the vertex, axis of symmetry, and roots (or zeros).

• The vertex is the highest or lowest point on the graph, depending on its opening direction.
• The axis of symmetry is a vertical line that passes through the vertex, splitting the parabola into mirror images.
• The roots are the points where the parabola intersects the x-axis, representing the solutions to the quadratic equation.

Quadratic functions are vital for understanding real-world phenomena in physics, engineering, and other scientific areas.

Horizontal Parabolas

Horizontal parabolas differ from their vertical counterparts as they open either left or right. This is indicated by equations in the form \((y - k)^2 = a(x - h)\). In this equation:

• \(h\) and \(k\) represent the coordinates of the vertex.
• The value of \(a\) determines the direction of the opening. If \(a > 0\), the parabola opens to the right. If \(a < 0\), it opens to the left.

For students, recognizing a horizontal parabola can be a bit tricky since \(y\) is squared instead of \(x\). This can lead to confusion if one is more used to the standard vertical forms. When tasked with analyzing or creating equations for horizontal parabolas, always focus on rearranging the terms to find clear insights about the direction and position of the parabola associated with the vertex coordinates.

Completing the Square

Completing the square is an essential algebraic skill used to transform quadratic equations into a more usable form, usually the vertex form. This technique is especially helpful when dealing with equations not easily factored or solved.
The main goal of completing the square is to form a perfect square trinomial on one side of the equation. Here's how you typically approach this:

• Start with a quadratic expression \(ax^2 + bx\).
• Divide \(b\) by 2 and square the result.
• Add and subtract this square inside the equation to maintain balance.
• Rearrange the equation so it forms a perfect square binomial. This often looks like \((x - h)^2\).

This structured method not only makes equations simpler to solve but also helps find the parabola's vertex, crucial information for graphing. Mastery of completing the square connects deeper algebraic skills to understanding quadratic forms and functions.

Vertex Form of a Parabola

The vertex form of a parabola is a powerful tool for understanding parabolas at a glance. This form is beneficial because it explicitly shows the vertex, making graphing much easier.
For a vertical parabola, the vertex form is \(y = a(x - h)^2 + k\), and for a horizontal parabola, it's \((y - k)^2 = a(x - h)\). In these equations:

• \((h, k)\) is the vertex of the parabola.
• The constant \(a\) reveals whether the parabola opens up or down (or left or right for horizontal parabolas).
• A positive \(a\) opens upwards/right, while a negative \(a\) opens downwards/left.

Converting an equation to vertex form often involves completing the square. Once in this form, the vertex can be identified quickly, which simplifies graph plotting. Knowing the vertex form highlights the parabola's symmetry and balance, providing clearer insights into its geometric properties.