Problem 35 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=-3(x-2)^{2}-1$$

Step-by-Step Solution

Answer

Plot the vertex (2, -1), the y-intercept (0, -13), and points (1, -4) and (3, -4) to draw the parabola.

Step 1: Identify the function form

The function given is written in vertex form: \(f(x) = a(x-h)^2 + k\). For our function, \(a = -3\), \(h = 2\), and \(k = -1\). The vertex form easily provides the vertex of the parabola at \((h, k)\).

Step 2: Determine the vertex

Identify the vertex from the function \(f(x) = -3(x-2)^2-1\). Here, the vertex is \((2, -1)\). This means the parabola opens downwards, as \(a\) is negative.

Step 3: Determine the axis of symmetry

The axis of symmetry can be identified from the vertex in the equation form, it is \(x = h\). In this case, \(x = 2\). This axis will divide the parabola into two symmetrical halves.

Step 4: Find y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(f(x)\).\[f(0) = -3(0-2)^2 - 1 = -3(4) - 1 = -12 - 1 = -13\]. So, the y-intercept is at \((0, -13)\).

Step 5: Find additional points

To obtain additional points, choose values of \(x\) near the vertex to substitute into the function. For example, calculate \(f(1)\) and \(f(3)\). \[ f(1) = -3(1-2)^2 - 1 = -3(1) - 1 = -4 \] and \[ f(3) = -3(3-2)^2 - 1 = -4 \]. Points are \((1,-4)\) and \((3,-4)\).

Step 6: Plot the points and draw the graph

Plot all the calculated points: the vertex \((2, -1)\), y-intercept \((0, -13)\), and additional points \((1, -4)\) and \((3, -4)\). Use the symmetry along \(x = 2\) to ensure the parabola is drawn accurately.

Key Concepts

Vertex Form

The vertex form of a quadratic function is a very user-friendly way to express these functions and makes graphing them more straightforward. It is given by the equation:

• \( f(x) = a(x-h)^2 + k \)

The values \( h \) and \( k \) in this equation primarily help us identify the vertex of the parabola, which is the peak or the lowest point on the graph.

Let's consider the function \( f(x) = -3(x-2)^2 - 1 \). By comparing this with the general vertex form, we conclude:

• The \( a \) coefficient is \( -3 \), indicating how the graph is stretched or compressed.
• \( h = 2 \) and \( k = -1 \), meaning our vertex is located at the point \((2, -1)\).

The negative \(a\) also informs us that the parabola opens downwards, making the vertex a maximum point. Understanding the vertex form not only reveals this crucial intersection but also aids in predicting how the parabola behaves without necessarily calculating multiple points.

Parabola Symmetry

A parabola is symmetric, meaning one side mirrors the other, which simplifies graphing significantly. Symmetry plays a crucial role because if we know one point on one side of the parabola, we automatically know a corresponding point on the other side.

In a visually symmetric parabola, the line of symmetry, also called the axis of symmetry, divides the parabola into two equal parts. For the function \( f(x) = -3(x-2)^2 - 1 \), this symmetry line means if you plot a point at \((1, -4)\) on one side, a point at \((3, -4)\) will be on the other side.

• The pairing occurs because both points are equally distant from the line of symmetry.
• This feature ensures that our parabola is shaped evenly, which is helpful in determining and verifying additional points.

Using this symmetry, you can assess whether you have plotted the points correctly and help to draw the parabola accurately, ensuring it retains its proper arch and direction.

Axis of Symmetry

The axis of symmetry is an imaginary vertical line that divides the parabola into two symmetrical halves.

This line always passes through the vertex and is defined mathematically by the equation \( x = h \).

• For our function with vertex \((2, -1)\), the axis of symmetry is \( x = 2 \).

This means the curve is perfectly mirrored across this axis in terms of shape and positioning of points.

To graph this, imagine drawing a vertical line through the vertex.

• Any two points lowering or rising the same distance from the axis have the same y-value.
• Knowing this, consistency on either side of the axis can be checked and maintained as you plot the parabola on a graph.

Understanding the axis of symmetry helps not only in graphing but also in quickly estimating values and transformations of the graph, providing a clearer visual representation of the quadratic function's behavior.

Y-intercept

The y-intercept of a function is where the graph crosses the y-axis. This occurs when the value of \( x \) is zero. Finding the y-intercept can add another helpful point to our graph.

For the quadratic function \( f(x) = -3(x-2)^2 - 1 \), calculate the y-intercept by substituting \( x = 0 \) into the equation.

• \[ f(0) = -3(0-2)^2 - 1 = -3(4) - 1 = -13 \]
• Thus, our y-intercept is \((0, -13)\).

This point is essential as it gives us an anchor on the vertical axis to orient the graph correctly.

In the context of already knowing the vertex and line of symmetry, the y-intercept helps verify that the graph is accurate. If plotted correctly, this intercept should maintain the parabola's symmetry around the axis that has been established during the graphing process. Bringing together all these aspects solidifies the understanding of how each plays into the quadratic function's visualization.