Problem 3 - Basic Technical Mathematics with Calculus

Question

Sketch the graph of each parabola by using only the vertex and the \(y\) -intercept. Check the graph using a calculator. \(y=x^{2}-6 x+5\)

Step-by-Step Solution

Answer

Vertex: (3, -4); y-intercept: (0, 5); the parabola opens upwards.

Step 1: Identify the Coefficients

The given quadratic equation is in the standard form \(y = ax^2 + bx + c\). Identify the coefficients: \(a=1\), \(b=-6\), and \(c=5\).

Step 2: Find the Vertex

The x-coordinate of the vertex can be found using the formula \(x = -\frac{b}{2a}\). Substitute \(b = -6\) and \(a = 1\) into this formula: \(x = -\frac{-6}{2 \times 1} = 3\). To find the y-coordinate of the vertex, substitute \(x = 3\) back into the equation: \(y = (3)^2 - 6(3) + 5 = 9 - 18 + 5 = -4\). Therefore, the vertex is \((3, -4)\).

Step 3: Find the y-intercept

The y-intercept occurs when \(x = 0\). Substitute \(x = 0\) into the equation: \(y = (0)^2 - 6(0) + 5 = 5\). Thus, the y-intercept is \((0, 5)\).

Step 4: Sketch the Parabola

Plot the points for the vertex \((3, -4)\) and the y-intercept \((0, 5)\). Since the parabola opens upwards (as \(a = 1 > 0\)), sketch a smooth curve through these points that is symmetric around the vertical line \(x = 3\), which is the axis of symmetry.

Step 5: Verify the Graph

Use a graphing calculator to input the equation \(y = x^2 - 6x + 5\) and compare the plotted graph with the sketch. Ensure the vertex and y-intercept match the calculated points.

Key Concepts

Quadratic equations

Quadratic equations are a type of polynomial equation with one variable raised to the second power. They are generally structured in the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable.
They form a parabola when graphed on a coordinate plane. The coefficient \(a\) determines the direction of the opening of the parabola:

• If \(a > 0\), the parabola opens upward, creating a U-shape.
• If \(a < 0\), it opens downward, forming an upside-down U.

The values of \(b\) and \(c\) influence the shape and placement of the parabola on the graph. Quadratic equations are a crucial part of algebra, enabling us to model a variety of real-world situations, from projectile motions to area problems.

Vertex of a parabola

The vertex of a parabola is its highest or lowest point, depending on whether it opens upward or downward. It provides critical information about the parabola's minimum or maximum value.
To find the vertex, use the formula for the x-coordinate: \(x = -\frac{b}{2a}\). This formula simplifies the process by giving us the axis of symmetry. Once we have the x-coordinate, the y-coordinate can be determined by substituting this x-value back into the original equation.
For example, in the equation \(y = x^2 - 6x + 5\), we find:

• The x-coordinate of the vertex: \(x = -\frac{-6}{2 \times 1} = 3\).
• Substitute \(x = 3\) into the equation to find the y-coordinate, yielding \(y = -4\).

Thus, the vertex is \((3, -4)\). Knowing the vertex helps in sketching the parabola accurately.

Y-intercept

The y-intercept is the point where the graph of an equation crosses the y-axis, which occurs when \(x = 0\). It is a key feature of the graph and provides a starting point for sketching.To find the y-intercept of a parabola, substitute \(x = 0\) into the quadratic equation. In our example of \(y = x^2 - 6x + 5\), we perform the calculation:

• \(y = (0)^2 - 6(0) + 5 = 5\)

The y-intercept, therefore, is the point \((0, 5)\). This point is easy to plot and, along with the vertex, helps us to better visualize and sketch the parabola's curve.

Graphing calculator

A graphing calculator is a valuable tool for visualizing complex mathematical equations, especially in verifying and sketching graphs of quadratic equations.
To use a graphing calculator effectively, enter the quadratic equation exactly as it appears: \(y = x^2 - 6x + 5\). After inputting the function, the calculator will plot the corresponding parabola.
Check the displayed graph against the manually calculated vertex and y-intercept to ensure accuracy:

• Verify that the vertex \((3, -4)\) is correctly plotted.
• Ensure the y-intercept \((0, 5)\) is marked correctly.

Graphing calculators provide a visual confirmation of the analysis, reinforcing understanding and offering a quick check to ensure all manual work aligns with the graphic representation.