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

Step-by-Step Solution

Answer

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

Step 1: Identify the Form of the Equation

The given equation is in the standard form of a quadratic equation, which is \(y = ax^2 + bx + c\). Here, \(a = 2\), \(b = -4\), and \(c = 5\). This form is useful for finding the vertex and axis of symmetry.

Step 2: Find the Vertex

The vertex of a parabola in standard form can be found using the formula \(x = -\frac{b}{2a}\). Substitute \(b = -4\) and \(a = 2\) into the formula: \[ x = -\frac{-4}{2 \cdot 2} = 1 \]Then, substitute \(x = 1\) back into the equation to find \(y\):\[ y = 2(1)^2 - 4 \cdot 1 + 5 = 3 \]Thus, the vertex is \((1, 3)\).

Step 3: Determine the Axis of Symmetry

The axis of symmetry for a parabola in standard form is the vertical line \(x = -\frac{b}{2a}\). From the previous step, we have \(x = 1\). So, the axis of symmetry is the line \(x = 1\).

Step 4: Determine the Domain

The domain of any quadratic function is all real numbers, as there are no restrictions on \(x\) values for a parabola. Therefore, the domain is \((-\infty, \infty)\).

Step 5: Determine the Range

Since the parabola opens upwards (because \(a = 2 > 0\)), the range starts from the y-coordinate of the vertex and goes to positive infinity. The vertex y-coordinate is 3, so the range is \([3, \infty)\).

Step 6: Sketch the Parabola

Plot the vertex at \((1, 3)\) on graph paper. Use the axis of symmetry to ensure the parabola is symmetric. Calculate additional points for accuracy if necessary and plot them. Draw a smooth curve to form the parabola.

Step 7: Verify with a Graphing Calculator

Input the function \(y = 2x^2 - 4x + 5\) into a graphing calculator and ensure the vertex, axis, domain, and range match the manual calculation and sketch. If the parabola matches the description (vertex at \((1, 3)\), axis of symmetry \(x=1\), etc.), the solution has been verified.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key point that represents either the lowest point (for parabolas that open upwards) or the highest point (for those that open downwards). It's like the tip or the peak of the parabola. In the context of a quadratic function in standard form, which is given by the equation \(y = ax^2 + bx + c\), the vertex can be found using a specific formula.

• To find the x-coordinate of the vertex, use the formula \(x = -\frac{b}{2a}\).
• After calculating the x-value, substitute it back into the original equation to find the y-coordinate.
• These coordinates \((x, y)\) provide the precise location of the vertex on the graph.

For the given quadratic function \(y = 2x^2 - 4x + 5\), substituting the values of \(a = 2\) and \(b = -4\) into the formula provides us with the vertex at \((1, 3)\). It plays a crucial part in graphing the parabola and determining other features such as the axis of symmetry and the range.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex, ensuring symmetry, and helps while sketching the parabola as each side is a reflection of the other.

In a quadratic function expressed in standard form, the axis of symmetry can be easily determined as it is given by the formula \(x = -\frac{b}{2a}\). This is the same formula used to find the x-coordinate of the vertex. Thus, finding the vertex inherently provides the axis of symmetry as well.

• The formula \(x = -\frac{b}{2a}\) is crucial as it demonstrates the dependency of the parabola on coefficients \(b\) and \(a\).
• For the quadratic function \(y = 2x^2 - 4x + 5\), applying this concept shows that the axis of symmetry is \(x = 1\).

Understanding the axis of symmetry can simplify the process of plotting additional points on either side of the parabola for a more accurate graph.

Domain and Range

The domain and range of a quadratic function are essential to understanding the extent of its graph.

- **Domain** refers to all possible input values (x-values) for the function. For any quadratic function, no matter its form, the domain is always all real numbers, \((-\infty, \infty)\). This is because you can substitute any real number into the function to get a corresponding y-value.- **Range**, on the other hand, refers to all possible output values (y-values). Unlike the domain, the range is influenced by the direction in which the parabola opens.
In our example \(y = 2x^2 - 4x + 5\), since the coefficient \(a = 2\) is positive, the parabola opens upwards. Therefore, the minimum y-value starts at the y-coordinate of the vertex, which is 3, and extends upwards to positive infinity. Thus, the range of this function is \([3, \infty)\).

Understanding the range is crucial for grasping how far the parabola extends vertically on the graph.