Problem 36 - Algebra for College Students

Question

Find the vertex for the graph of each quadratic function. $$y=3 x^{2}-2 x+1$$

Step-by-Step Solution

Answer

The vertex is at (1/3, 2/3).

Step 1: Identify the coefficients

In the quadratic function y = 3x^2 - 2x + 1the coefficients are: a = 3b = -2c = 1.

Step 2: Use the vertex formula

The x-coordinate of the vertex for a quadratic function in the form y = ax^2 + bx + c is given by: x = -b/(2a) Substitute the values of a and b:

Step 3: Calculate the x-coordinate

Calculate the x-coordinate of the vertex: x = -(-2)/(2*3) = 2/6 = 1/3.

Step 4: Calculate the y-coordinate

Substitute the x-coordinate back into the original quadratic function to find the y-coordinate: y = 3(1/3)^2 - 2(1/3) + 1 Simplify step-by-step: y = 3(1/9) - 2/3 + 1 y = 1/3 - 2/3 + 1 y = -1/3 + 3/3 y = 2/3.

Step 5: Write the vertex coordinates

The vertex of the quadratic function y = 3x^2 - 2x + 1 is (1/3, 2/3).

Key Concepts

quadratic equation

A quadratic equation is a type of polynomial equation of the second degree. The general form is given by: y = ax^2 + bx + c
Here, a, b, and c are constants, and the variable x is raised to the second power. Quadratic equations create parabolas when graphed.
Key Points:

• 'a' cannot be zero, as it would make the equation linear.
• The highest exponent of the variable x is 2.
• Depending on the value of 'a', the parabola can open upwards or downwards. If 'a' > 0, it opens upwards; if 'a' < 0, it opens downwards.

Quadratic equations often appear in physics, engineering, and many real-world scenarios. Understanding these equations provides a foundation for further study in mathematics.

vertex formula

The vertex formula is a crucial tool for finding the highest or lowest point of a parabola. The vertex is where the function changes direction. For a quadratic equation in the form y = ax^2 + bx + c, the x-coordinate of the vertex is given by the formula: x = -b/(2a)
Once you have the x-coordinate, you can find the corresponding y-coordinate by substituting the x value back into the original equation.
Steps to Find the Vertex:

• Identify coefficients 'a' and 'b' from the quadratic equation.
• Substitute 'a' and 'b' into the vertex formula x = -b/(2a) to calculate the x-coordinate.
• Substitute the x-coordinate back into the equation to find the y-coordinate.

Example Calculation:
Given the quadratic function y = 3x^2 - 2x + 1, identify 'a' = 3 and 'b' = -2.
Using the vertex formula: x = -(-2)/(2*3) = 1/3
Substitute x = 1/3 back into the equation to find y: y = 3(1/3)^2 - 2(1/3) + 1 = 2/3
The vertex is at (1/3, 2/3).

parabola

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

• The direction of opening depends on the coefficient 'a' in the quadratic equation. If 'a' > 0, it opens upwards. If 'a' < 0, it opens downwards.
• The vertex is the highest or lowest point on the parabola. This point can be found using the vertex formula.
• The axis of symmetry is a vertical line passing through the vertex, dividing the parabola into two mirror-image halves. This line has the equation x = h, where 'h' is the x-coordinate of the vertex.

Example: Consider the quadratic function y = 3x^2 - 2x + 1:
The parabola opens upwards because 'a' = 3 > 0.
The vertex, found previously, is at (1/3, 2/3).
The axis of symmetry is the line x = 1/3.
Understanding these features helps in graphing quadratic functions and analyzing their properties in different contexts.