Problem 25 - A Graphical Approach to College Algebra

Question

For each quadratic function defined , (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=x^{2}-10 x+21$$

Step-by-Step Solution

Answer

The vertex is (5, -4); graph is an upward parabola.

Step 1: Identify the Coefficients

For the quadratic function \(P(x) = x^2 - 10x + 21\), identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = 1\), \(b = -10\), and \(c = 21\).

Step 2: Use the Vertex Formula

The vertex form of a parabola is given by \(h = -\frac{b}{2a}\). Substitute the coefficients into this formula: \[ h = -\frac{-10}{2 \times 1} = \frac{10}{2} = 5 \]

Step 3: Calculate the y-coordinate of the Vertex

Substitute \(x = 5\) into the original function \(P(x)\) to find the y-coordinate of the vertex. \[ P(5) = (5)^2 - 10 \times 5 + 21 = 25 - 50 + 21 = -4 \]

Step 4: Write the Vertex Coordinates

The vertex of the function \(P(x) = x^2 - 10x + 21\) is \((5, -4)\).

Step 5: Sketch the Graph

Draw the parabola opening upwards (since \(a > 0\)) with the vertex at \((5, -4)\). Plot the vertex, and recall that it is symmetric about this point. Use the y-intercept, \((0, 21)\), and another point for accuracy.

Key Concepts

Vertex Formula

The vertex formula is a fundamental tool in analyzing quadratic functions. It helps you find the vertex of a parabola, which is a key point indicating the parabola's highest or lowest point, depending on its orientation. The vertex formula is given by \[ h = -\frac{b}{2a} \] where \( a \) and \( b \) are coefficients from the standard form of a quadratic equation:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)

To use the vertex formula, simply substitute these coefficients into the formula to find the \( x \)-coordinate of the vertex. This formula is excellent for determining the line of symmetry for the graph, as the vertex lies exactly halfway between the parabola's roots. Once you have \( h \), you can find the \( y \)-coordinate by substituting \( h \) back into the original quadratic equation.

Graphing Parabolas

Graphing parabolas involves plotting the key features of a quadratic function on a coordinate plane. The graph of a quadratic function is a U-shaped curve called a parabola. To accurately graph a parabola, you usually perform the following steps:

• Find the vertex using the vertex formula.
• Determine the way the parabola opens by looking at the sign of \( a \).
• Identify additional points like the y-intercept and other points on either side of the vertex to define its shape better.

After identifying these points, plot them on the graph. The vertex is your central point, and from there you can draw the symmetrical curve. Remember, if \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.

Parabola Vertex

The parabola vertex is a critical point of any quadratic graph. It represents the maximum or minimum point of the graph, dictated by whether the parabola opens upwards or downwards. Finding the coordinates of the vertex involves two main steps:1. Use the vertex formula to find the \( x \)-coordinate.2. Substitute this \( x \)-coordinate back into the original quadratic equation to get the \( y \)-coordinate.For example, using the function \( P(x) = x^2 - 10x + 21 \), apply the vertex formula to find the \( x \)-coordinate as 5. Substituting \( x = 5 \) into the function will then give you the \( y \)-coordinate of -4. Therefore, the parabola vertex is at \((5, -4)\). This vertex is a guide for graphing the function, helping to define the parabola's shape and position on a plot.

Y-Intercept

The y-intercept is the point where the graph of a function crosses the y-axis. For quadratic functions in the standard form \( ax^2 + bx + c \), the y-intercept can be easily found as it occurs at the point \((0, c)\). This is because when \( x = 0 \), the function simplifies to \( c \).In our example with \( P(x) = x^2 - 10x + 21 \), the y-intercept is \( (0, 21) \). The y-intercept is a useful reference when graphing a parabola, as it provides one more point, besides the vertex, for plotting the curve accurately.
Remember, the y-intercept provides insight into the parabola's height at the point where \( x = 0 \), which is particularly useful for quick sketching and understanding of the graph's overall behavior.