Problem 97 - A Graphical Approach to College Algebra

Question

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=x^{2}-2 x+3$$

Step-by-Step Solution

Answer

Vertex: (1, 2), Axis: x = 1, Domain: (-∞, ∞), Range: [2, ∞).

Step 1: Identify the Quadratic Function

The given equation is a quadratic function: \[ y = x^2 - 2x + 3 \] which is in the standard form \( y = ax^2 + bx + c \), with \( a=1 \), \( b=-2 \), and \( c=3 \).

Step 2: Calculate the Vertex

To find the vertex of the parabola, use the vertex formula:\[ x = -\frac{b}{2a} \]Substituting \( a = 1 \) and \( b = -2 \):\[ x = -\frac{-2}{2 \times 1} = 1 \]Plug this back into the equation to find \( y \):\[ y = (1)^2 - 2 \times 1 + 3 = 1 \times 1 - 2 + 3 = 2 \]Thus, the vertex of the parabola is at point \((1, 2)\).

Step 3: Determine the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. For this parabola, it is given by:\[ x = 1 \]

Step 4: Establish the Domain

For any quadratic function, the domain is all real numbers. Therefore, the domain of the function is:\[ (-\infty, \infty) \]

Step 5: Ascertain the Range

Since the parabola opens upwards (as \( a > 0 \)), the range starts from the \( y \)-coordinate of the vertex and goes to infinity:\[ [2, \infty) \]

Step 6: Graph the Parabola by Hand

Use the vertex \((1, 2)\) and additional points to draw the parabola. Calculate a few additional points by substituting \( x \) values into the equation:- For \( x = 0 \), \( y = 0^2 - 2 \times 0 + 3 = 3 \) - For \( x = 2 \), \( y = 2^2 - 2 \times 2 + 3 = 3 \)Plot these points: \((0, 3)\), \((1, 2)\), \((2, 3)\) and draw the symmetric curve.

Step 7: Verify Using a Graphing Calculator

Input the equation \( y = x^2 - 2x + 3 \) into the graphing calculator. Confirm the shape of the graph and check that the vertex \((1, 2)\), axis \( x = 1 \), and points \((0, 3)\) and \((2, 3)\) accurately match the hand-drawn graph.

Key Concepts

Vertex Calculation

The vertex of a parabola is a significant point because it represents the peak or the lowest tip of the graph. For the quadratic function given, it's essential to use the vertex formula to find this point. The vertex formula is given as: \[ x = -\frac{b}{2a} \]Here, \( a = 1 \) and \( b = -2 \). Substituting these values into the formula gives:\[ x = -\frac{-2}{2 \times 1} = 1 \]This calculation gives us the x-coordinate of the vertex. To find the y-coordinate, substitute \( x = 1 \) back into the original equation:\[ y = (1)^2 - 2 \times 1 + 3 = 2 \]Thus, the vertex is at the point \((1, 2)\). It's always a good practice to check this point is located accurately on the graph, as it helps guide the drawing of the entire parabola.

Axis of Symmetry

The axis of symmetry is an imaginary line that divides the parabola into two mirror-image halves. It always passes through the vertex of the parabola. For every quadratic function, the formula for the axis of symmetry can be directly derived from the x-coordinate of the vertex. For our quadratic function, after calculating the vertex, we found that the x-coordinate is 1. Therefore, the axis of symmetry is represented by the vertical line:\[ x = 1 \]This line of symmetry helps in plotting the points on the parabola since any point on one side will have a corresponding point on the other side at the same distance from this line. Remember, the axis of symmetry is vital when sketching the graph of a parabola because it ensures that the graph is drawn accurately and symmetrically.

Domain and Range

The domain and range are fundamental concepts in understanding the behavior of quadratic functions and their graphs.

• Domain: For any quadratic function, the domain is all real numbers since the parabola extends horizontally without any restriction. This can be expressed as:\[ (-\infty, \infty) \]
• Range: The range of a quadratic function is determined by the direction the parabola opens. Since our quadratic function has a positive leading coefficient \( a = 1 \), the parabola opens upwards. The lowest point on the graph is the vertex, with a y-coordinate of 2. Therefore, the range consists of all numbers starting from this lowest point going to infinity:\[ [2, \infty) \]

By understanding the domain and range, you can determine the extent of your graph both horizontally and vertically, which provides a comprehensive view of the quadratic function's behavior.

Quadratic Functions

A quadratic function is a second-degree polynomial function characterized by its equation of the form:\[ y = ax^2 + bx + c \]Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). The graph of a quadratic function is a parabola. Depending on the value of \( a \), the parabola can open upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)). In terms of our function \( y = x^2 - 2x + 3 \), \( a = 1 \) which means this parabola opens upwards. Key features of quadratic functions include:

• The vertex, which indicates the maximum or minimum point of the parabola.
• The axis of symmetry, which passes through the vertex and acts as a line of reflection for the graph.
• The domain and range, which describe the set of possible input and output values of the function.

Graphically, these features assist in plotting the parabola accurately by hand or validating it with a graphing calculator. Understanding these aspects ensures a solid grasp of how quadratic functions behave in different scenarios.