Problem 47 - Beginning and Intermediate Algebra

Question

Graph each function using the vertex formula. Include the intercepts. \(g(x)=2 x^{2}-4 x+4\)

Step-by-Step Solution

Answer

The vertex of the parabola is V(1, 2), the x-intercept is (1, 0) coinciding with the vertex, and the y-intercept is (0, 4). To graph the function \(g(x) = 2x^2 - 4x + 4\), plot these points and sketch an upward-opening parabola.

Step 1: Find the vertex

The vertex formula is given by \[x = \frac{-b}{2a}\], where a and b are the coefficients in the quadratic function \(g(x) = ax^2 + bx + c\). In this case, \(a = 2\) and \(b = -4\). Plugging these values into the formula to find the x-coordinate of the vertex, we get \[x = \frac{-(-4)}{2(2)} = \frac{4}{4} = 1\]. To find the y-coordinate of the vertex, plug the x-coordinate back into the function: \[g(1) = 2(1)^2 - 4(1) + 4 = 2 - 4 + 4 = 2\]. Therefore, the vertex of the parabola is V(1, 2).

Step 2: Find the intercepts

To find the x-intercepts, we set the function equal to zero and solve for x: \[0 = 2x^2 - 4x + 4\]. However, it's difficult to factor this equation or solve for x directly. So, let's use the quadratic formula to find the x-intercepts: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. With \(a = 2\), \(b = -4\), and \(c = 4\), we get: \[x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(4)}}{2(2)} = \frac{4 \pm \sqrt{0}}{4}\]. Since the discriminant is 0 (\(b^2 - 4ac = 0\)), there is only one x-intercept: \[x = \frac{4}{4} = 1\]. This means the x-intercept coincides with the vertex, and the point is (1,0). To find the y-intercept, we set \(x = 0\) and solve for \(g(x)\): \[g(0) = 2(0)^2 - 4(0) + 4 = 4\]. This gives us the y-intercept at the point (0, 4).

Step 3: Plot the points and sketch the parabola

Now we have the vertex V(1, 2), the x-intercept (1, 0), and the y-intercept (0, 4). Plot these points on a graph, and then sketch the parabola using these points, remembering that the graph opens upwards (since the coefficient \(a = 2 > 0\)). This will give you the graph of the function \(g(x) = 2x^2 - 4x + 4\).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function with the highest degree of 2. It is generally expressed in the form \(g(x) = ax^2 + bx + c\), where:

• \(a\), \(b\), and \(c\) are constants
• \(a\) is not equal to zero (otherwise it wouldn't be a quadratic)

The "axt^2" term makes the function a parabola when plotted on a graph.
The key characteristic of a quadratic function is its curve shape, which is symmetrical and can open upwards or downwards.

• If \(a\) is positive, the parabola opens upwards.
• If \(a\) is negative, it opens downwards.

Quadratics are invaluable in various applications, including physics and engineering, where they model trajectories and other real-world phenomena.

Parabola Graphing

Graphing a parabola is an essential skill when dealing with quadratic functions.
A parabola is a U-shaped curve that reflects the graph of a quadratic equation. The most critical point in a parabola is the vertex, which is its highest or lowest point depending on its orientation.
In the equation \(g(x) = 2x^2 - 4x + 4\):

• The vertex was found at (1,2), where the curve reaches its minimum.
• The vertex can be found using the vertex formula \(x = \frac{-b}{2a}\)

Knowing where the vertex is helps in sketching the symmetry of the parabola. Other important points include the intercepts, which provide more structure to our graph.

Intercepts

Intercepts are points where the graph crosses the x-axis or y-axis.

• **X-intercepts** are points where \(g(x) = 0\). They represent where the graph meets the x-axis.
• **Y-intercept** is the point where \(x = 0\) and the graph meets the y-axis.

In the exercise:

• We found the y-intercept at (0,4) by substituting \(x = 0\), yielding \(g(0) = 4\).
• The x-intercept coincided with the vertex at (1,0), because the discriminant (\(b^2 - 4ac\)) equaled zero. This means there was only one x-intercept.

Recognizing these intercepts is an integral part of effectively sketching the parabola.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation, specifically pointing out where the function touches or crosses the x-axis (x-intercepts).
It is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, the values \(a\), \(b\), and \(c\) are the coefficients from the quadratic function \(ax^2 + bx + c\). This solution mechanism is invaluable, especially when factoring is difficult or impossible. In our lesson:

• We used the quadratic formula and found that the discriminant \(b^2 - 4ac\) was zero. This indicated that the quadratic equation has exactly one solution.
• This single solution was corroborated by the fact that the vertex and x-intercept coincided.

The quadratic formula thus not only helps graph a quadratic function accurately but also reveals insights about the nature of its graph.