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

The vertex is \((3, -4)\) and the y-intercept is \((0, 5)\).

Step 1: Find the Vertex

To find the vertex of the parabola, we use the formula for the x-coordinate of the vertex, given by \(-\frac{b}{2a}\), where \(a\) and \(b\) are the coefficients from the quadratic equation \(y=ax^2+bx+c\). Here, \(a=1\), \(b=-6\), and \(c=5\). Thus, the x-coordinate is \(x = -\frac{-6}{2 \times 1} = 3\). Substitute \(x = 3\) back into the equation to find the y-coordinate: \(y = (3)^2 - 6 \times 3 + 5 = 9 - 18 + 5 = -4\). Thus, the vertex is at the point \((3, -4)\).

Step 2: Find the Y-intercept

The y-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the equation: \(y = (0)^2 - 6 \times 0 + 5 = 5\). Therefore, the y-intercept is at the point \((0, 5)\).

Step 3: Sketch the Graph

Using the vertex \((3, -4)\) and the y-intercept \((0, 5)\), we can sketch the parabola. Start by plotting these two points on a graph. Since the coefficient of \(x^2\) is positive, the parabola opens upwards. Draw a smooth, symmetrical curve through these points, ensuring that the vertex is the lowest point on the curve and the parabola passes through the y-intercept.

Step 4: Confirm with Calculator

Use a graphing calculator to enter the equation \(y=x^2-6x+5\). Check to ensure the calculator's graph matches the sketch. Confirm that the vertex and y-intercept align with the plotted points \((3, -4)\) and \((0, 5)\), respectively.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key point that indicates the highest or lowest point on the graph of a quadratic function. In other words, it is the point where the graph changes direction. For a quadratic equation in the format \(y = ax^2 + bx + c\), the x-coordinate of the vertex can be found using the formula \(x = -\frac{b}{2a}\). This formula is derived from completing the square and helps identify the symmetry axis of the parabola.

• The value of \(x\) in the vertex provides the line of symmetry for the parabola.
• The y-coordinate is found by substituting the x-coordinate back into the original equation.

In our given equation \(y = x^2 - 6x + 5\), substituting \(a = 1\) and \(b = -6\) into the formula gives \(x = 3\). Substituting \(x = 3\) back into the equation, we find the corresponding y-value as \(-4\), giving the vertex \((3, -4)\). This point is crucial as it tells us that the parabola reaches its lowest point here, as the parabola opens upwards due to the positive \(a\) value.

Y-intercept

The y-intercept of a quadratic function is where the graph crosses the y-axis. This occurs at the point where the input, or \(x\), is zero. To find this intercept, simply substitute \(x = 0\) into the quadratic equation and solve for \(y\).

• The y-intercept gives a direct point on the graph that is easy to plot.
• It also helps to confirm the shape and direction of the parabola when sketching by hand.

For our equation \(y = x^2 - 6x + 5\), substituting \(x = 0\) results in \(y = 5\), creating the point \((0, 5)\). By marking this point on the graph, it serves as a guide to ensure the parabola's curve is accurate and intersects the y-axis correctly. It supports the sketch, emphasizing the symmetry and the general shape influenced by the vertex.

Quadratic Functions

Quadratic functions represent equations based on the format \(y = ax^2 + bx + c\). These functions generate a parabolic graph, offering several important characteristics like the vertex and y-intercept. Understanding these concepts is crucial for sketching and interpreting the graphs of quadratic functions.

• The coefficient \(a\) decides the parabola's opening direction (upward if positive, downward if negative).
• The vertex provides the turning point, allowing us to identify the function's maximum or minimum value.
• The y-intercept offers an easy-to-locate reference point.

Graphing quadratics involves using these features to sketch the curve. In the case of \(y = x^2 - 6x + 5\), the function is simplified by plotting known points like the vertex \((3, -4)\) and y-intercept \((0, 5)\). Align the symmetry of the parabola with these points to create an accurate representation. For more accuracy, a graphing calculator can validate the sketch by displaying the full parabola. This enhances understanding and ensures the graph accurately reflects the function's behavior.