Problem 42 - Algebra A Combined Function

Question

Find the vertex of the graph of each function. $$ h(x)=x^{2}-3 $$

Step-by-Step Solution

Answer

The vertex of \(h(x) = x^2 - 3\) is \((0, -3)\).

Step 1: Identify the Function Type

The function given is in the form of a quadratic function, which is generally expressed as \(y = ax^2 + bx + c\). In this case, \(h(x) = x^2 - 3\), where \(a = 1\), \(b = 0\), and \(c = -3\).

Step 2: Determine the Formula for the Vertex

For a quadratic function in the standard form \(y = ax^2 + bx + c\), the vertex's x-coordinate can be found using the formula \(x = -\frac{b}{2a}\). Since \(b = 0\), the formula simplifies to \(x = 0\).

Step 3: Calculate the y-coordinate of the Vertex

Substitute \(x = 0\) back into the function to find the y-coordinate of the vertex: \(h(0) = (0)^2 - 3 = -3\).

Step 4: Write the Vertex in Coordinate Form

The vertex of the function \(h(x) = x^2 - 3\) is \((0, -3)\). This means that the vertex lies on the y-axis and is 3 units below the origin.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree of the variable is 2. Quadratic functions are commonly written in the form of \(y = ax^2 + bx + c\). Here:

• \(a\), \(b\), and \(c\) are constants.
• \(x\) is the variable.
• The value of \(a\) determines the direction of the parabola -- upwards if \(a > 0\) and downwards if \(a < 0\).

Quadratic functions graph as parabolas, which are U-shaped curves. These functions are widely used in various fields such as physics, engineering, and economics to model situations where variables change in a non-linear manner, like projectile motion or profit/loss calculations. Understanding the key aspects of quadratic functions is foundational for exploring more complex mathematical concepts.

Standard Form

The standard form of a quadratic function is expressed as \(y = ax^2 + bx + c\). Each component in this equation plays a crucial role:

• \(a\) is the leading coefficient, which affects the width and direction of the parabola.
• \(b\) is the linear coefficient, influencing the parabola's horizontal placement and orientation.
• \(c\) is the constant term, representing the y-intercept of the graph.

Finding the vertex and other features of the parabola is easier when the equation is in standard form. Additionally, the standard form provides straightforward ways to execute operations like factoring or completing the square, which can further assist in sketching or transforming the graph.

Parabola Vertex

The vertex of a parabola is the peak or lowest point of the curve, depending on the direction it opens. For the quadratic function \(y = ax^2 + bx + c\), the vertex can be determined using the formula for the x-coordinate: \(x = -\frac{b}{2a}\).

• If \(b = 0\), the vertex is simply \((0, c)\), where all changes depend on the constant term.
• The vertex represents a point of symmetry for the parabola.
• For \(h(x)=x^2-3\), substituting \(b=0\) and \(a=1\) gives the vertex as \((0, -3)\).

Understanding the vertex is essential, as it provides valuable insight into the function's maximum or minimum value and its graph's overall shape.

Coordinate Form

Coordinate form involves writing the position of specific points on a graph using an ordered pair \((x, y)\). This is particularly useful when identifying or describing the location of significant points like the vertex.

• The ordered pair consists of an x-coordinate (horizontal position) and a y-coordinate (vertical position).
• The vertex in coordinate form for \(h(x)=x^2-3\) is \((0, -3)\), indicating the vertex lies directly on the y-axis at \(y = -3\).
• This notation provides a clear and concise way to visualize and communicate points on a graph.

Using coordinate form is critical in graphing functions, solving equations, and conveying mathematical ideas efficiently, aiding in clearer problem-solving and analysis.