Problem 99 - 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. $$y=2 x^{2}-4 x+5$$

Step-by-Step Solution

Answer

Vertex: (1, 3), Axis: x = 1, Domain: (-∞, ∞), Range: [3, ∞).

Step 1: Identify the Quadratic Form

The given equation is in the form of a quadratic equation: \( y = ax^2 + bx + c \) where \( a = 2 \), \( b = -4 \), and \( c = 5 \). This form is suitable for determining various properties of the parabola.

Step 2: Find the Vertex

The vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Calculate \( x = -\frac{-4}{2 \times 2} = 1 \). Substitute \( x = 1 \) back into the equation \( y = 2(1)^2 - 4(1) + 5 \) to find \( y = 3 \). Thus, the vertex is \((1, 3)\).

Step 3: Determine the Axis of Symmetry

The axis of symmetry for a parabola with a known vertex \((h, k)\) is \( x = h \). Therefore, since the vertex is \((1, 3)\), the axis of symmetry is \( x = 1 \).

Step 4: Define the Domain

For any quadratic function of the form \( y = ax^2 + bx + c \), the domain is all real numbers. Thus, the domain is \((-fty, fty)\).

Step 5: Define the Range

Since \( a = 2 \) is positive, the parabola opens upwards. The minimum value of \( y \) is the \( y\)-coordinate of the vertex, which is \( 3 \). Therefore, the range of \( y \) is \([3, fty)\).

Step 6: Plot the Parabola

To sketch the graph, plot the vertex at \((1, 3)\) on the graph and draw the axis of symmetry along \( x = 1 \). Since \( a = 2 \), which is positive, the parabola opens upward. Mark a few more points on the graph by choosing values for \( x \) (like 0, 2, 3) and calculating their corresponding \( y \) values to ensure the parabolic shape is accurate.

Step 7: Verify Using a Graphing Calculator

Input the equation \( y = 2x^2 - 4x + 5 \) into a graphing calculator. Check that the vertex is at \((1, 3)\), the axis of symmetry is \( x = 1 \), and the parabola opens upwards as expected with the correct domain and range.

Key Concepts

Quadratic Equation

In mathematics, a quadratic equation is any equation that can be rewritten in the form:

• \( y = ax^2 + bx + c \)

where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). A quadratic equation graphically represents a parabola on the Cartesian plane. The shape of the parabola depends on the values of these constants.

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

In our specific example of \( y = 2x^2 - 4x + 5 \), we identify it as a quadratic equation with \( a = 2 \), \( b = -4 \), and \( c = 5 \). This configuration implies the parabola opens upwards, as \( a \) is positive, creating a U-shaped curve.

Vertex

The vertex of a parabola is a critical point that represents the minimum or maximum of the quadratic function. For a parabola of the form \( y = ax^2 + bx + c \), the vertex can be calculated using the vertex formula:

• \( x = -\frac{b}{2a} \)

Once the \( x \)-coordinate is found, substitute it back into the original quadratic equation to find the \( y \)-coordinate.
In our example with \( y = 2x^2 - 4x + 5 \):

• Calculate \( x = -\frac{-4}{2 \times 2} = 1 \).
• Find \( y \) by substituting \( x = 1 \) back into the equation: \( y = 2(1)^2 - 4(1) + 5 = 3 \).

Therefore, the vertex of the parabola is \((1, 3)\). This point is crucial as it provides a guide for sketching the parabola.

Axis of Symmetry

The axis of symmetry of a parabola is an imaginary vertical line that divides the parabola into two symmetrical halves. This line passes through the vertex. For a parabola described by \( y = ax^2 + bx + c \), the axis of symmetry is given by the formula:

• \( x = -\frac{b}{2a} \)

In our example, this means:

• With a vertex at \((1, 3)\), the axis of symmetry is \( x = 1 \).

This axis aids in determining that each side of the parabola mirrors the other, providing a balanced graph and helping to predict the position of reflected points on the curve.

Domain and Range

The domain and range describe the set of possible inputs (\( x \) values) and outputs (\( y \) values) for a function, respectively. For any quadratic function,

• The domain is always all real numbers \((-\infty, \infty)\).

This is because parabolas extend indefinitely in both directions along the \( x \)-axis.
However, the range of the parabola depends on the direction in which it opens. For the quadratic \( y = 2x^2 - 4x + 5 \):

• The parabola opens upward, indicating all \( y \) values starting from the vertex \( y \)-coordinate.
• Here, the vertex is \((1, 3)\), so the range is \([3, \infty)\).

Understanding the domain and range helps further in sketching graphs and predicting behavior of the quadratic function. It's a foundational concept for analyzing function graphs.