Problem 56 - Algebra for College Students

Question

Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=(x-20)^{2}$$

Step-by-Step Solution

Answer

Plot the vertex at (20, 0), draw the axis of symmetry at x = 20, and plot additional points to sketch the parabola opening upwards.

Step 1 - Identify the form of the polynomial

The given polynomial function is \( f(x) = (x-20)^2 \). This is a quadratic function in the vertex form \( f(x) = (x-h)^2 \) where \( h = 20 \) and it opens upwards.

Step 2 - Identify the vertex of the polynomial

The vertex form of the quadratic function indicates that the vertex is located at the point \( (h, 0) \) or \( (20, 0) \).

Step 3 - Determine the axis of symmetry

The axis of symmetry for a quadratic in this form is a vertical line that passes through the vertex. Thus, the axis of symmetry is \( x = 20 \).

Step 4 - Plot the vertex and axis of symmetry

On graph paper or using a graphing calculator, plot the vertex at \( (20, 0) \). Then, draw the vertical line \( x = 20 \) which is the axis of symmetry.

Step 5 - Plot additional points

To sketch the graph more accurately, choose a few x-values around the vertex and calculate corresponding y-values. For example: \( x = 19 \rightarrow f(19) = (19-20)^2 = 1 \) and \( x = 21 \rightarrow f(21) = (21-20)^2 = 1 \). Plot these points as well: \( (19, 1) \) and \( (21, 1) \).

Step 6 - Sketch the parabola

Using the vertex, axis of symmetry, and additional points, sketch the parabola. It should be a U-shaped curve opening upwards with the vertex at \( (20, 0) \) and symmetric about the line \( x = 20 \).

Key Concepts

quadratic functions

Quadratic functions are a fundamental concept in algebra. They take the form of a polynomial of degree 2, typically written as \( f(x) = ax^2 + bx + c \). These functions graph into a shape called a parabola. The main feature of quadratic functions is their symmetry and specific vertex point, making them valuable in various mathematical applications.
Some key points include:

• The highest degree term is always squared.
• The graph is a U-shaped curve (a parabola).
• They open upward if \(a > 0 \) and downward if \(a < 0 \).

Understanding quadratic functions is crucial as they appear in many real-world scenarios such as physics, engineering, and economics.

vertex form

Vertex form is a special way to express quadratic functions. It makes identifying the vertex of the parabola straightforward. A quadratic in vertex form looks like \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex:

• \( a \) indicates the parabola's openness and direction.
• \( h \) is the x-coordinate of the vertex.
• \( k \) is the y-coordinate of the vertex.

For example, in the exercise's function \( f(x) = (x-20)^2 \), the vertex form is clear:

• The vertex is at \( (20, 0) \).
• The parabola opens upwards, as the coefficient is positive.
• There is no \( k \) value, indicating the vertex lies on the x-axis.

Using the vertex form helps in plotting quadratic functions accurately and understanding their geometric properties.

parabola

A parabola is the graph of a quadratic function, characterized by its U-shape and symmetry. All parabolas share common features whether they are opening upwards or downwards:

• Vertex: The highest or lowest point of the parabola.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
• Direction: Determined by the leading coefficient (\( a \)); positive \( a \) opens upward, negative opens downward.
• Focus and Directrix: Specific points associated with parabolas used in more advanced studies.

In the exercise function \( f(x) = (x-20)^2 \):

• The vertex is at \( (20, 0) \).
• The axis of symmetry is the line \( x = 20 \).
• It opens upwards, forming a clear U-shape with reliable symmetry for easy graphing.

Understanding parabolas allows students to predict and draw quadratic graphs efficiently.