Problem 10 - Technical Mathematics with Calculus

Question

Graph each function. Set the viewing window for \(x\) and \(y\) initially from -5 to 5 then resize if needed. $$y=3.73-1.77 x^{2}$$

Step-by-Step Solution

Answer

The graph is a downward opening parabola with vertex at (0, 3.73) and axis of symmetry at \(x=0\). The viewing window is set from -5 to 5 for both \(x\) and \(y\), and should be adjusted if the parabola extends beyond this window.

Step 1: Recognize the type of function

Identify that the given function is a quadratic function in the form of \(y=ax^2+bx+c\), where \(a=-1.77\), \(b=0\), and \(c=3.73\). This function represents a parabola that opens downwards because \(a<0\).

Step 2: Find the vertex of the parabola

The vertex of a parabola in the form of \(y=ax^2+bx+c\) is given by the point \((h,k)\), where \(h=-\frac{b}{2a}\) and \(k=c-\frac{b^2}{4a}\). Since \(b=0\), the vertex is at \((0,c)\), or \((0,3.73)\).

Step 3: Plot the vertex

Begin plotting the function by placing a point at the vertex \((0,3.73)\) on the graph.

Step 4: Find the axis of symmetry

The axis of symmetry is a vertical line that passes through the vertex. For the given function, the axis of symmetry is \(x=0\).

Step 5: Choose x-values and calculate y-values

Select \(x\)-values around the vertex to calculate corresponding \(y\)-values. For example, choose x-values like -1, 0, and 1.

Step 6: Plot additional points

Using the chosen \(x\)-values, calculate other points using the given function, such as \((1, 3.73-1.77(1^2))\) and \((-1, 3.73-1.77(-1^2))\). Plot these points on the graph.

Step 7: Draw the parabola

With the vertex and additional points plotted, draw a smooth curve to form the parabola.

Step 8: Adjust the viewing window if necessary

Initially set the viewing window from -5 to 5 for both \(x\) and \(y\), then adjust the window if the shape of the parabola is not clear or if it does not fit within the current viewing window.

Key Concepts

Parabola Characteristics

A parabola is a type of curve on a graph commonly represented by a quadratic function of the form \(y = ax^2 + bx + c\). It features a distinctive U-shaped curve that can open either upwards or downwards depending on the sign of \(a\). When \(a > 0\), the parabola opens upwards, and when \(a < 0\), it opens downwards.

For the function \(y = 3.73 - 1.77x^2\), we identify that the parabola opens downwards due to the negative coefficient of the \(x^2\) term, which is \(a = -1.77\). This downward opening indicates that the parabola has a maximum point at its vertex. Other notable characteristics include its symmetry about a central axis, known as the axis of symmetry, and its vertex, the highest or lowest point on the graph.

Vertex of a Parabola

The vertex of a parabola is a crucial concept in graphing as it represents the peak or trough of the curve, which is either the maximum or minimum point. For the quadratic function \(y = ax^2 + bx + c\), the vertex can be found by calculating \(h = -\frac{b}{2a}\) and \(k = c - \frac{b^2}{4a}\).

The function \(y = 3.73 - 1.77x^2\) simplifies this process as \(b = 0\), making \(h = 0\) and \(k \)= the constant term, which is 3.73. Therefore, the vertex of this parabola is at the point \( (0, 3.73)\), and since the parabola opens downwards, this vertex is the maximum point. The vertex serves as a helpful starting point for graphing the parabola, as it aids in understanding the function's overall shape and direction.

Axis of Symmetry

The axis of symmetry in a parabolic graph is a vertical line that divides the parabola into two mirror images. For a quadratic function, this axis of symmetry always passes through the vertex of the parabola. Mathematically, the axis of symmetry is defined by the equation \(x = h\), where \(h\) is the \(x\)-coordinate of the vertex of the parabola.

For the function given, since the vertex is at \( (0, 3.73)\), the axis of symmetry is the line \(x = 0\). This means that for any point \( (x, y)\) on the parabola, there will be a corresponding point \( (-x, y)\) that is equidistant from the axis of symmetry. The axis of symmetry is not just a visual aid but also an important feature used to reflect points across the parabola when graphing or to analyze the function's behavior.