Problem 30 - Algebra A Combined Function

Question

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ y=x^{2}+4 $$

Step-by-Step Solution

Answer

The graph is a parabola with vertex at (0, 4).

Step 1: Recognize the Equation Type

The given equation is \( y = x^2 + 4 \). This equation is in the form \( y = ax^2 + bx + c \), which is the standard form of a quadratic equation in terms of \( y \). Quadratic equations represent parabolas.

Step 2: Identify the Key Features of the Parabola

The equation \( y = x^2 + 4 \) is a parabola that opens upwards since the coefficient of \( x^2 \) is positive (i.e., \( 1 \)). The vertex of a parabola in the form \( y = ax^2 + c \) is at the point \((0, c)\). Here, the vertex is at \( (0, 4) \).

Step 3: Sketch and Label the Graph

To sketch the graph, plot the vertex at \((0, 4)\). Draw a symmetric curve opening upwards from this vertex. The vertex is the highest or lowest point, depending on the parabola's orientation. In this case, label the vertex \( V(0, 4) \) on the graph.

Key Concepts

Parabola

A parabola is a unique, U-shaped curve that you often encounter in mathematics. It is one of the simplest forms of conic sections and is the graph of a quadratic function. The distinctive feature of a parabola is its symmetry. It means that if you fold it along a particular line (the axis of symmetry), both halves will match perfectly.

In the most common form, a parabola is described by the equation

• \( y = ax^2 + bx + c \)

The orientation of the parabola (whether it opens upwards or downwards) depends on the sign of the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards, resembling a smile.
• If \( a < 0 \), it opens downwards, like a frown.

Parabolas appear in various real-life scenarios, such as the path of a ball thrown in the air or the design of certain satellite dishes. Understanding them is a key step in mastering quadratic equations.

Quadratic Equation

Quadratic equations form the backbone of understanding parabolas. They are polynomial equations of the second degree, usually written as

• \( ax^2 + bx + c = 0 \)

Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). This equation sets the stage for numerous concepts in algebra and calculus.

The solutions to a quadratic equation give us the x-values where the graph intersects the x-axis. They're typically found using methods such as:

• Factoring the quadratic expression.
• Completing the square.
• Using the quadratic formula: \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Each solution method has its advantages and practical use cases. Quadratic equations play a critical role in modeling situations in physics, engineering, and many other fields.

Vertex of a Parabola

The vertex is a crucial concept when dealing with parabolas. It is either the highest or lowest point on the graph, depending on its orientation. In simpler terms, it is the point where the parabola changes direction.

For a parabola given by the equation

• \( y = ax^2 + bx + c \)

the vertex can be found using the formula:

• Vertex \((h, k)\) where \( h = \frac{-b}{2a} \) and \( k = c - \frac{b^2}{4a} \).

In special cases like \( y = ax^2 + c \), the vertex is easily identifiable because there is no \( b \) term, and it sits at \((0, c)\).

The vertex represents the most extreme point of the parabola:

• For a parabola opening upwards, it is the minimum point.
• For a parabola opening downwards, it is the maximum point.

This feature is valuable for solving optimization problems in various contexts, particularly in economics and physics, where finding maximum and minimum values is crucial.