Problem 44 - Precalculus : Building Concepts and Connections

Question

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$f(x)=3 x^{2}-12 x+4$$

Step-by-Step Solution

Answer

The vertex of the parabola is \((2, -4)\), the axis of symmetry is \(x = 2\), and two additional points on the parabola are \((1, -5)\) and \((3, -5)\).

Step 1: Find the Vertex

The vertex \((h, k)\) of a parabola can be found using the formula \(h = -\frac{b}{2a}\), and \(k = f(h)\). In the provided equation \(f(x)=3x^{2}-12x+4\), the coefficients are \(a = 3\), \(b = -12\) and \(c = 4\). Therefore, to find \(h\), substitute the values of \(a\) and \(b\) into the formula, giving \(h = -\frac{-12}{2*3} = 2\). To find \(k\), substitute \(h = 2\) into the function to get \(k = 3(2)^{2}-12(2)+4 = -4\). Therefore the vertex is \((2,-4)\).

Step 2: Determine the Axis of Symmetry

The axis of symmetry of a parabola is the vertical line passing through the vertex, i.e. \(x = h\). Therefore, for the given function, the axis of symmetry is \(x = 2\).

Step 3: Find Additional Points on the Parabola

To find additional points, substitute different \(x\) values into the function and compute their corresponding \(f(x)\) or \(y\) values. Let's choose \(x = 1\) and \(x = 3\), for example. If \(x = 1\), then \(f(x) = 3(1)^{2}-12(1)+4 = -5\). So, one point on the parabola is \((1, -5)\). If \(x = 3\), then \(f(x) = 3(3)^{2}-12(3)+4 = -5\). So, the second point is \((3, -5)\).

Step 4: Sketch the Parabola

The sketching will be based on the vertex, the axis of symmetry, and the two additional points. However, given that this is a text-based response, graphical representation is not possible here but can be done manually on a piece of paper or using any graphic software.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point where the direction of the curve changes. In simpler terms, it is the highest or lowest point of the parabola, depending on its orientation. For a given quadratic function in the form \( f(x) = ax^2 + bx + c \), the vertex can be calculated using the formula \( h = -\frac{b}{2a} \) for the x-coordinate. The corresponding y-coordinate, \( k \), is found by plugging the x-value into the function, that is, \( k = f(h) \).
For example, in the function \( f(x) = 3x^2 - 12x + 4 \), \( a = 3 \), \( b = -12 \), and \( c = 4 \). Using the formula, we calculate \( h = 2 \) as shown in the solution. Plugging this back into the function gives \( k = -4 \). So, the vertex of the parabola is \((2, -4)\). This point is important because it gives the peak or trough of the parabola.

• Identify the coefficients \( a, b, \) and \( c \).
• Calculate \( h \) using \( h = -\frac{b}{2a} \).
• Find \( k \) by evaluating \( f(h) \).

Remember, the vertex deeply affects the shape and position of the parabola within the coordinate system.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror images. It always passes through the vertex. In relation to its equation, this line is described as \(x = h\), where \(h\) is the x-coordinate of the vertex.
For the quadratic function \( f(x) = 3x^2 - 12x + 4 \), we've already determined the vertex to be \((2, -4)\). Hence, the axis of symmetry is the line \(x = 2\). This line is important because it helps in determining the shape and orientation of the parabola. Knowing this line makes it easier to sketch the parabola as this line is a guide for graphical representation.

• Find the x-coordinate of the vertex.
• Write the equation of the line as \(x = h\).

The axis of symmetry is significant as it tells us that for every point on one side of the axis, there is a corresponding point directly opposite it on the other side.

Quadratic Function

A quadratic function is a polynomial function with a degree of 2, typically expressed in the form \( f(x) = ax^2 + bx + c \). It graphically represents a parabola. The value of \( a \) determines the direction of the parabola (upward or downward) and its width.
If \( a > 0 \), the parabola opens upwards, creating a U-shape. If \( a < 0 \), it opens downwards. The larger the absolute value of \( a \), the narrower the parabola.
The quadratic function derives its remarkable properties from this equation:

• The vertex, calculated as discussed, affects where the highest or lowest point of the parabola will be.
• The axis of symmetry \( x = h \) divides it into two equal halves.
• The roots or x-intercepts, if they exist, are the points where the function crosses the x-axis.

Understanding quadratic functions is crucial because they are foundational in algebra and have numerous applications in science, engineering, and beyond. They model many real-world scenarios where maximum or minimum points are of interest.