Problem 34 - College Algebra with Corequisite Support

Question

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts. $$ f(x)=x^{2}-2 x $$

Step-by-Step Solution

Answer

Vertex: (1, -1), Axis of symmetry: x = 1, Intercepts: (0, 0) and (2, 0).

Step 1: Identify the Quadratic Function Form

The given quadratic function is \( f(x) = x^2 - 2x \). This is in the standard form \( ax^2 + bx + c \) where \( a = 1 \), \( b = -2 \), and \( c = 0 \).

Step 2: Find the Vertex

The vertex of a quadratic function \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). Substituting the values, we have \( x = -\frac{-2}{2(1)} = 1 \). Plugging \( x = 1 \) back into the function to get \( y \): \( y = (1)^2 - 2(1) = -1 \). Thus, the vertex is \( (1, -1) \).

Step 3: Determine the Axis of Symmetry

The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is a vertical line through the vertex, given by \( x = -\frac{b}{2a} \). From the previous step, we know the axis of symmetry is \( x = 1 \).

Step 4: Find the Y-intercept

The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) in the function, we have \( f(0) = 0^2 - 2(0) = 0 \). Therefore, the y-intercept is \( (0, 0) \).

Step 5: Find the X-intercepts

The x-intercepts occur when \( f(x) = 0 \). Solving the equation \( x^2 - 2x = 0 \), we factor it as \( x(x - 2) = 0 \), giving solutions \( x = 0 \) or \( x = 2 \). Thus, the x-intercepts are \( (0, 0) \) and \( (2, 0) \).

Step 6: Sketch the Graph

Using the information from previous steps, sketch the graph of \( f(x) = x^2 - 2x \). Plot the vertex \( (1, -1) \), the axis of symmetry \( x = 1 \), the y-intercept \( (0, 0) \), and the x-intercepts \( (0, 0) \) and \( (2, 0) \). Draw the parabolic curve opening upwards, passing through these points.

Key Concepts

Vertex

Quadratic functions are often characterized by their distinctive U-shaped graphs known as parabolas. At the heart of this parabola lies the vertex. The vertex represents the peak or the lowest point of the curve, depending on its orientation. In our quadratic function, given by \( f(x) = x^2 - 2x \), you can locate this vertex by using a simple formula.

To find the vertex, use the formula \( x = -\frac{b}{2a} \). In our function, \( a \) is 1, and \( b \) is -2. Plugging these values into the formula gives \( x = 1 \). Once we know the x-coordinate, we substitute it back into the function to find the y-coordinate:

• \( y = (1)^2 - 2(1) = -1 \)

Thus, the vertex is located at the point \( (1, -1) \). Understanding the vertex is crucial because it provides insights into the maximum or minimum value of the function, showing where the curve turns.

Axis of Symmetry

A unique feature of parabolic graphs in quadratic functions is the axis of symmetry. This is an imaginary vertical line that precisely divides the parabola into two mirrored halves. For any quadratic equation \( ax^2 + bx + c \), the axis of symmetry can be easily calculated using the same formula for finding the x-coordinate of the vertex. Once again, it’s given by \( x = -\frac{b}{2a} \).

In the case of our function \( f(x) = x^2 - 2x \), the axis of symmetry occurs at \( x = 1 \).

• The line \( x = 1 \) passes vertically through the vertex \( (1, -1) \).
• It ensures that the parabola is symmetric, reflecting changes on one side of the axis to the other side.

This symmetry is important for graphing purposes, helping us predict the behaviour of the parabola on either side of the axis.

Intercepts

Intercepts are key points where the graph crosses the axes, providing vital information at a glance. Let's explore the x-intercepts and the y-intercept for the function \( f(x) = x^2 - 2x \).

**Y-intercept**
The y-intercept is found where the graph crosses the y-axis, which occurs when \( x = 0 \). At \( x = 0 \):

• \( f(0) = 0^2 - 2(0) = 0 \)

This means the y-intercept is at the point \( (0, 0) \).

**X-intercepts**
These are points where the graph crosses the x-axis, found by setting the entire function equal to zero. Solve \( x^2 - 2x = 0 \) by factoring:

• Factor the equation as \( x(x - 2) = 0 \)
• This gives solutions \( x = 0 \) or \( x = 2 \)

Consequently, the x-intercepts are at points \( (0, 0) \) and \( (2, 0) \). Recognizing these intercepts aids in sketching the graph accurately, ensuring you capture where it interacts with the axes.