Problem 14 - Beginning and Intermediate Algebra

Question

Identify the vertex of each parabola. $$ f(x)=(x+5)^{2}-8 $$

Step-by-Step Solution

Answer

The vertex is \((-5, -8)\).

Step 1 - Recognize the Vertex Form of a Parabola

The given function is in the form of \(f(x) = (x-h)^2 + k\), which represents a parabola. In this case, the vertex form makes it easy to identify the vertex.

Step 2 - Identify Parameters

Compare the given function \(f(x) = (x+5)^2 -8\) to the standard form \(f(x) = (x-h)^2 + k\). This shows that \(h = -5\) and \(k = -8\).

Step 3 - State the Vertex

The vertex \((h, k)\) is thus \((-5, -8)\).

Key Concepts

Vertex Form of a Parabola

When dealing with quadratic functions, the vertex form is extremely useful. It is written as \(f(x) = (x - h)^2 + k\). This form clearly shows the vertex of the parabola.

The vertex is the point \((h, k)\), which is the highest or lowest point on the graph depending on the direction of the parabola. If the coefficient of the squared term is positive, the parabola opens upwards, and if it's negative, the parabola opens downwards.
Understanding the vertex form helps us quickly identify the vertex, which is crucial in graphing and solving quadratic equations.

Understanding Parabolas

A parabola is a U-shaped curve that can open either upwards or downwards. It is the graph of a quadratic function.

Key characteristics of parabolas include:

• The vertex is the highest or lowest point.
• The axis of symmetry is a vertical line through the vertex.
• The direction the parabola opens is determined by the sign of the coefficient in the quadratic term.

Parabolas are everywhere in the real world, from the trajectory of a thrown ball to the design of satellite dishes. Recognizing their properties allows us to analyze and understand these natural phenomena better.

Parameters Identification

In the vertex form \(f(x) = (x - h)^2 + k\), identifying the parameters \(h\) and \(k\) is essential.

Comparing with the given function \(f(x) = (x+5)^2 -8\), we can see:

• \(h = -5\) because \(x + 5\) can be written as \(x - (-5)\).
• \(k = -8\), falling right after the quadratic term.

These steps make finding the vertex straightforward. Just identify these parameters and the vertex is ready to be stated.

Coordinate Geometry in Parabolas

Coordinate geometry allows us to analyze and graph quadratic functions as parabolas on the Cartesian plane. By using vertex form \((x - h)^2 + k\), we can easily locate the vertex \((h, k)\). Here's how:

• Plot the vertex on the coordinate plane.
• Draw the axis of symmetry through the vertex.
• Plot additional points if needed for a precise graph.

Coordinate geometry helps visualize the relationships between algebraic expressions and their graphical representations, making understanding quadratic functions easier and more intuitive.