Problem 78 - Algebra and Trigonometry

Question

Explaining the Concepts Describe how to find a parabola's vertex if its equation is in the form \(f(x)=a x^{2}+b x+c .\) Use \(f(x)=x^{2}-6 x+8\) as an example.

Step-by-Step Solution

Answer

The vertex of the parabola represented by the equation \(f(x)=x^{2}-6 x+8\) is (3, -1).

Step 1: Identify the coefficients a, b and c

From the standard form \(f(x)=ax^{2}+bx+c\), identify the coefficients 'a', 'b' and 'c'. For the example equation \(f(x)=x^{2}-6x+8\), the coefficient 'a' is 1, 'b' is -6 and 'c' is 8.

Step 2: Find the x-coordinate of the vertex

The x-coordinate of the vertex can be found by the formula \(-b/2a\). Substituting 'a' and 'b' with their respective values gives: \(x=-(-6)/(2*1)=3\). So, the x-coordinate of the vertex is 3.

Step 3: Find the y-coordinate of the vertex

Substitute x=3 in the given quadratic equation to find the y-coordinate. So, \(f(3)=3^{2}-6*3+8=-1\). Hence the y-coordinate of the vertex is -1.

Key Concepts

Vertex Formula

The vertex of a parabola provides a crucial point that indicates the maximum or minimum value of the quadratic function. Understanding how to find the vertex using the vertex formula is essential in graphing parabolas. The vertex itself is a point,

• denoted as \((h, k)\)
• where the coordinates can be found using specific calculations.

For a given quadratic equation in the form \(f(x) = ax^2 + bx + c\), we can find the vertex using the formula for the x-coordinate: \(-\frac{b}{2a}\).
First, determine this x-coordinate by plugging the values of 'a' and 'b' into \(-\frac{b}{2a}\),simplifying to find a specific numerical value.
Once the x-coordinate is identified, substitute it back into the original equation to find the y-coordinate. Together, these values
form the complete vertex.

• Example: If \(f(x) = x^2 - 6x + 8\), the vertex formula helps us find that \(x = 3\), then substituting back provides \(y = -1\).

Parabola

A parabola is a U-shaped curve that represents the graphical output of a quadratic function. Its properties can vary:

• It can open upwards or downwards depending on the value of coefficient 'a'.

When 'a' is positive, the parabola opens upward, forming a smile shape; when 'a' is negative, it opens downward, resembling a frown. The vertex of the parabola is a pivotal feature, as it represents the highest or lowest point on the curve.
Understanding the direction of opening and the vertex of a parabola is fundamental in sketching the graph of a quadratic equation. Alongside the axis of symmetry, which runs vertically through the vertex, these elements help define the shape and position of the parabola in a coordinate plane.

Standard Form

The standard form of a quadratic equation is \(f(x) = ax^2 + bx + c\), where:

• 'a' is the leading coefficient,
• 'b' is the linear coefficient,
• and 'c' is the constant term.

This form is particularly useful for identifying the coefficients, which are crucial in various calculations:

• It allows us to apply the vertex formula easily to find the vertex.

By standardizing the way equations are written, it becomes simpler to transform them or compare them against one another.
When solving problems, recognizing an equation in standard form enables swift identification of key components necessary for graphing or further algebraic manipulation.

Coefficients

Coefficients are numbers that multiply the variables or products in an equation. Their values hint at how a particular quadratic behaves:

• The leading coefficient 'a' determines the openness and direction of the parabola.
• The linear coefficient 'b' affects the position of the vertex horizontally.
• The constant 'c' gives insight into the parabola's vertical position on the graph.

For example, in the quadratic equation \(f(x) = x^2 - 6x + 8\),
'a' is 1, which means the parabola opens upward due to the positive sign,'b' is -6, and'c' is 8, setting the y-intercept at \((0, 8)\).
Accurately identifying these coefficients allows for proper application of the vertex formula and guides overall graphing techniques.