Problem 94 - A Graphical Approach to Precalculus with Limits

Question

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=(x-5)^{2}-4$$

Step-by-Step Solution

Answer

Vertex: (5, -4), Axis: x = 5, Domain: (-∞, ∞), Range: [-4, ∞).

Step 1: Identify the Vertex

The equation of the parabola is given in the vertex form: \( y = (x - h)^2 + k \), where \( (h, k) \) is the vertex. For the equation \( y = (x-5)^2 - 4 \), the vertex is \((5, -4)\).

Step 2: Determine the Axis of Symmetry

The axis of symmetry for a parabola in vertex form \( y = (x - h)^2 + k \) is the vertical line \( x = h \). For our equation, the axis of symmetry is \( x = 5 \).

Step 3: Find the Domain

The domain of a quadratic function, like this parabola, is all real numbers. In interval notation, the domain is \((-\infty, \infty)\).

Step 4: Find the Range

The range of the parabola depends on whether it opens upwards or downwards. Since the coefficient of \((x-5)^2\) is positive, the parabola opens upwards. The minimum value of \( y \) is at the vertex, which is \(-4\). Therefore, the range is \([-4, \infty)\).

Step 5: Sketch the Parabola by Hand

Start by plotting the vertex \((5, -4)\) on the graph. Then, draw the axis of symmetry at \( x = 5 \). Since the parabola opens upwards, sketch a U-shaped curve starting from the vertex, ensuring that the curve is symmetric about the axis of symmetry.

Step 6: Verify Using a Graphing Calculator

Input the equation \( y = (x-5)^2 - 4 \) into the graphing calculator. Check that the graph on the calculator matches the sketch you have drawn by hand, with vertex at \( (5, -4) \), axis of symmetry \( x = 5 \), and the parabola opening upwards.

Key Concepts

Vertex Form

The vertex form of a quadratic equation is a powerful tool for analyzing and graphing parabolas. This form is expressed as \( y = (x-h)^2 + k \). Here, the coordinates \((h, k)\) serve as the vertex of the parabola, which is the turning point or the point where the parabola changes direction.
In our exercise, the equation is \( y = (x-5)^2 - 4 \). This format clearly shows that the vertex is \((5, -4)\). This means the parabola will have its lowest point at \(x = 5\) and \(y = -4\).
Using the vertex form allows for easy identification of the vertex, which is crucial for graphing the parabola accurately. It highlights how these simple transformations from the basic \( y = x^2 \) shape adjust the position of the curve.

Axis of Symmetry

An important characteristic of a parabola is its axis of symmetry. This is an imaginary vertical line that divides the parabola into two equal mirror-image halves.
For a parabola in vertex form \( y = (x - h)^2 + k \), the axis of symmetry is found at \( x = h \). In the equation provided, \( y = (x-5)^2 - 4 \), the axis of symmetry is \( x = 5 \).
This axis runs right through the vertex, ensuring that for every point \( (x, y) \) on one side of the axis, there is a corresponding point on the opposite side at \( (2h - x, y) \). Understanding the axis of symmetry makes it easier to graph the parabola by ensuring balance and symmetry around this central line.

Domain and Range

The domain and range of a parabola describe the set of possible inputs (x-values) and outputs (y-values) the function can take on.

• The domain of any parabola with a quadratic equation is all real numbers, or \((-\ )\infty, \infty)\). This is because the curve extends infinitely left and right on the x-axis.
• The range of a parabola, however, depends on whether it opens upwards or downwards. Since our equation \( y = (x-5)^2 - 4 \) opens upwards (the coefficient of the square term is positive), the minimum point is the y-coordinate of the vertex. Thus, the range is \([-4, \infty)\). This indicates that the smallest y-value is \(-4\), and as x moves away from the vertex, y can become infinitely large.

Knowing the domain and range not only guides you in sketching the parabola correctly but is essential in understanding the possible values for the function.