Problem 27 - 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. $$y=-x^{2}+4 x-2$$

Step-by-Step Solution

Answer

The vertex is (2, 2) and the parabola opens downward.

Step 1: Identify the coefficients

The given quadratic function is \( y = -x^2 + 4x - 2 \). Identify the coefficients: \( a = -1 \), \( b = 4 \), and \( c = -2 \). These coefficients will be used in the vertex formula.

Step 2: Use the vertex formula

The vertex formula for a quadratic function in standard form \( y = ax^2 + bx + c \) is \( x = -\frac{b}{2a} \). Substitute the values of \( a \) and \( b \) into the formula: \( x = -\frac{4}{2(-1)} = \frac{4}{2} = 2 \).

Step 3: Find the y-coordinate of the vertex

Use the \( x \)-coordinate from Step 2 to find the \( y \)-coordinate of the vertex by substituting \( x = 2 \) back into the function: \( y = -(2)^2 + 4(2) - 2 = -4 + 8 - 2 = 2 \). Thus, the vertex is at \((2, 2)\).

Step 4: Graph the function

Start by plotting the vertex we found, \((2, 2)\). Since the coefficient of \( x^2 \) is negative (\( a = -1 \)), the parabola opens downward. Identify that the axis of symmetry is \( x = 2 \). Next, find additional points by choosing \( x \) values around the vertex to see the shape of the parabola. For example, when \( x = 1 \), \( y = -(1)^2 + 4(1) - 2 = 1 \), and when \( x = 3 \), \( y = -(3)^2 + 4(3) - 2 = -2 \). Plot these points accordingly.

Key Concepts

Vertex Formula

A quadratic function often appears in the form of \( y = ax^2 + bx + c \). The vertex of the parabola, a crucial point representing its peak or valley, can be determined using the **vertex formula**. This formula gives us the x-coordinate of the vertex: \( x = -\frac{b}{2a} \). In this formula, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
Substitute the values of \(a = -1\) and \(b = 4\) from our example function \( y = -x^2 + 4x - 2 \).
The vertex x-coordinate is calculated as:
\[ x = -\frac{4}{2(-1)} = \frac{4}{2} = 2 \]
Once the x-coordinate is known, the y-coordinate can be found by substituting back into the function:
\[y = -(2)^2 + 4(2) - 2 = -4 + 8 - 2 = 2 \]
So, the vertex is located at the point \((2, 2)\). This point plays a key role in sketching the parabola, providing a powerful starting point for further analysis.

Graphing Parabolas

Once we have identified the vertex, graphing a parabola becomes more manageable.
The vertex, here at \((2, 2)\), is critical for understanding the parabola's behavior, such as its direction of opening. Because the coefficient of \(x^2\) (\(a = -1\)) is negative, this parabola opens downwards. Think of the vertex as the peak of a hill when \(a\) is negative.
Here's a step-by-step guide to graphing parabolas:

• Plot the vertex on the coordinate plane.
• The **axis of symmetry** assists in symmetry and runs vertically through the vertex. Our axis here is \(x = 2\).
• Choose x-values around the vertex to calculate additional points (e.g., \( x = 1, 3 \)). Find their respective y-values.
• Connect the points in a smooth curve to form the parabola.

Knowing these steps fosters a deeper understanding and visual representation of quadratic functions.

Quadratic Coefficients

Quadratic functions are described by the coefficients \(a\), \(b\), and \(c\) in the expression \( y = ax^2 + bx + c \). These coefficients offer insights into the parabola's properties.
Here's what the coefficients mean:

• **Coefficient \(a\):** Determines the direction and width of the parabola.
  • If \(a > 0\), the parabola opens upwards.
  • If \(a < 0\), it opens downwards, like in our example, where \(a = -1\).
  • The greater the absolute value of \(a\), the narrower the parabola.
• **Coefficient \(b\):** Together with \(a\), impacts the x-value of the vertex. It influences the slope of the tangent line at the vertex.
• **Coefficient \(c\):** Represents the y-intercept, the point where the graph intersects the y-axis.

By understanding these coefficients, solving quadratic expressions for both analytical calculations and practical graphing becomes much clearer.