Problem 5 - Algebra A Combined Function

Question

Find the vertex of the graph of each quadratic function by completing the square or using the vertex formula. $$ f(x)=5 x^{2}-10 x+3 $$

Step-by-Step Solution

Answer

The vertex of the quadratic function is (1, -2).

Step 1: Identify the form of the quadratic equation

The given quadratic function is \( f(x) = 5x^2 - 10x + 3 \). This is in the standard form \( ax^2 + bx + c \), where \( a = 5 \), \( b = -10 \), and \( c = 3 \).

Step 2: Use the Vertex Formula

The vertex of a parabola of form \( ax^2 + bx + c \) can be found using the vertex formula \( x = -\frac{b}{2a} \). Substitute \( a = 5 \) and \( b = -10 \) into the formula.

Step 3: Calculate the x-coordinate of the vertex

Plug in the values into the formula: \( x = -\frac{-10}{2\times 5} = \frac{10}{10} = 1 \). The x-coordinate of the vertex is 1.

Step 4: Find the y-coordinate of the vertex

Substitute \( x = 1 \) back into the original function to find the y-coordinate: \( f(1) = 5(1)^2 - 10(1) + 3 = 5 - 10 + 3 = -2 \).

Step 5: Write the vertex in coordinate form

The vertex of the function, expressed as a coordinate, is \( (1, -2) \).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that is crucial in mathematics because it represents parabolas when graphed. It is generally written in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The graph of a quadratic function is a curve known as a parabola.

Some interesting features of quadratic functions include:

• The degree of the function is 2, which means the highest power of \( x \) is \( x^2 \).
• The graph of a quadratic function is symmetric.
• The sign of \( a \) determines whether the parabola opens upwards \((a > 0)\) or downwards \((a < 0)\).
• The vertex of the parabola represents the minimum point if \( a > 0 \), or the maximum point if \( a < 0 \).

Understanding these properties helps in analyzing quadratic functions visually and algebraically. Knowing the role each parameter (\( a \), \( b \), and \( c \)) plays can make it easier to predict and understand how changing these values affects the graph.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, making it easier to determine characteristics of its graph, such as the vertex. This involves creating an expression that looks like \((x-h)^2 + k\), where \( (h, k) \) is the vertex of the parabola.

Here's how you can complete the square for a quadratic function:

• Start with the standard quadratic form \( ax^2 + bx + c \).
• Factor out \( a \) from the first two terms, if \( a eq 1 \). This gives you \( a(x^2 + \frac{b}{a}x) + c \).
• Within the parentheses, add and subtract \( (\frac{b}{2a})^2 \) to create a perfect square trinomial.
• Rewrite the expression as \( a(x-h)^2 + k \) where \( h = \frac{-b}{2a} \) and \( k \) is adjusted to complete the equation.

This method not only simplifies finding the vertex but also allows for easy graphing of the quadratic function. By converting it into the vertex form, it's simpler to see how shifts in \( h \) and \( k \) translate to the movement of the parabola on the graph.

Parabola Vertex

The vertex of a parabola is a significant point that defines its position and orientation, acting as either the lowest or highest point of the graph. For any quadratic function of the form \( ax^2 + bx + c \), the vertex serves as a pivot point that highlights the symmetry of the parabola.

To find the vertex, one can use the vertex formula \( x = -\frac{b}{2a} \), which calculates the x-coordinate of the vertex. By substituting this x-value back into the original quadratic equation, the corresponding y-coordinate can be found.

For example, given the quadratic function \( f(x) = 5x^2 - 10x + 3 \):

• Use \( x = -\frac{-10}{2 \times 5} = 1 \) to find the x-coordinate.
• Substitute \( x = 1 \) back into the equation: \( f(1) = 5(1)^2 - 10(1) + 3 = -2 \), making the vertex \( (1, -2) \).

The vertex provides valuable insights into the parabolic graph, indicating either the peak or the trough based on the direction the parabola opens. Being able to quickly find and understand the vertex adds efficiency to solving many algebraic and real-world problems.