Problem 9 - Beginning and Intermediate Algebra

Question

Identify the vertex of each parabola. $$ f(x)=x^{2}+4 $$

Step-by-Step Solution

Answer

The vertex of the parabola is (0, 4).

Step 1: Understanding the vertex form of a parabola

The general form of a quadratic function is given by \[ f(x) = ax^2 + bx + c \]However, the vertex form of a quadratic function is \[ f(x) = a(x-h)^2 + k \]where ( h, k) is the vertex of the parabola.

Step 2: Compare the given function to the vertex form

The given function is \[ f(x) = x^2 + 4 \]By comparing it to the vertex form \[ f(x) = a(x-h)^2 + k \]we can see that there is no (x-h) term in our function, meaning h = 0 and k = 4.

Step 3: Identify the vertex

From the comparison in Step 2, it can be concluded that the vertex ( h, k) is (0, 4).Hence, the vertex of the given parabola is at the point (0, 4).

Key Concepts

quadratic function

A quadratic function is a type of polynomial function specifically for degree 2 equations. This means any equation where the highest power of the variable, usually denoted as x, is squared. The standard form of a quadratic function is represented as \( f(x) = ax^2 + bx + c \). Quadratic functions are important in algebra because they describe the shape of a parabola when graphed.

Some key features of a quadratic function include:

• The coefficient 'a' determines the direction of the parabola. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards.
• The highest or lowest point of the parabola is called the vertex.
• The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror images.
• The parabola may or may not intersect the x-axis, and at most it can intersect at two points called the roots or solutions.

Understanding how quadratic equations work is fundamental to mastering parabolas and solving related problems, such as finding the vertex.

vertex form

The vertex form of a quadratic function is a way of writing the equation so that the vertex is easily identifiable. The vertex form is given by \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

This format is particularly useful in easily locating the vertex without performing complex calculations. In the vertex form:

• \