Problem 16 - Basic Technical Mathematics with Calculus

Question

Sketch the graph of each parabola by using the vertex, the \(y\) -intercept, and two other points, not including the \(x\) -intercepts. Check the graph using a calculator. $$y=-3 x^{2}-x$$

Step-by-Step Solution

Answer

The parabola has a vertex at \(\left(\frac{1}{6}, -\frac{1}{4}\right)\), y-intercept \((0,0)\), and opens downwards.

Step 1: Identify the Vertex

For a parabola in the form \(y = ax^2 + bx + c\), the vertex \((h,k)\) can be found using \(h = -\frac{b}{2a}\). Here, \(a = -3\) and \(b = -1\). Calculate \(h\): \(h = -\frac{-1}{2(-3)} = \frac{1}{6}\). To find \(k\), substitute \(h\) into the equation: \(k = -3\left(\frac{1}{6}\right)^2 - \frac{1}{6} = -\frac{1}{12} - \frac{1}{6} = -\frac{3}{12} = -\frac{1}{4}\). Thus, the vertex is \(\left(\frac{1}{6}, -\frac{1}{4}\right)\).

Step 2: Find the y-intercept

The y-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the equation: \(y = -3(0)^2 - 0 = 0\). So, the y-intercept is the point \((0, 0)\).

Step 3: Choose and Calculate Two Other Points

Select \(x = 1\) and \(x = -1\) to find two additional points on the parabola. For \(x = 1\), substitute into the equation: \(y = -3(1)^2 - 1 = -3 - 1 = -4\). So the point is \((1, -4)\). For \(x = -1\), \(y = -3(-1)^2 - (-1) = -3 + 1 = -2\). Thus the point is \((-1, -2)\).

Step 4: Sketch the Graph

Plot the vertex \(\left(\frac{1}{6}, -\frac{1}{4}\right)\), the y-intercept \((0, 0)\), and the points \((1, -4)\) and \((-1, -2)\) on the graph. The parabola opens downward because \(a = -3\) is negative. Draw a smooth curve through these points.

Step 5: Confirm Using a Calculator

Use a graphing calculator to verify the shape of the parabola. Input \(y = -3x^2 - x\) and check that the graph matches the sketched points and shape.

Key Concepts

Vertex Calculation

Calculating the vertex of a parabola is the first step to understanding its mysterious curves. The vertex is a unique point on the graph, often the peak or bottom of the parabola. It acts as a pivotal indicator of where the graph changes direction. To find the vertex of a function given by the quadratic form \( y = ax^2 + bx + c \), follow these steps:

• Calculate the x-coordinate \( h \) of the vertex using the formula: \( h = -\frac{b}{2a} \).
• Find the y-coordinate \( k \) by substituting \( h \) back into the original equation. Compute \( y = a(h)^2 + bh + c \) to find \( k \).

In our example, with \( a = -3 \) and \( b = -1 \), we calculated \( h = \frac{1}{6} \) and \( k = -\frac{1}{4} \), so the vertex is \( \left(\frac{1}{6}, -\frac{1}{4}\right) \). The vertex not only tells us about the parabola's highest or lowest point, but it is essential when graphing accurately.

Quadratic Function

The quadratic function forms the backbone of the beautiful parabolic curves we see in mathematics. It's expressed in the form \( y = ax^2 + bx + c \) and has distinctive properties. Here's why it matters:

• The quadratic function determines the shape of the parabola. The sign and value of \( a \) dictate whether it opens upwards or downwards.
• Parabolas can capture the paths of projectiles in physics or illustrate profit curves in economics.
• Understanding this function allows us to find key features like the vertex, axis of symmetry, and intercepts.

Our equation \( y = -3x^2 - x \) exemplifies these concepts:- The negative \( a \) value indicates the parabola opens downwards.- The intercepts provide initial points of contact with the axes, vital when plotting the graph. Embracing the quadratic function means unraveling the secrets behind its elegant curve.

Graphing Techniques

Graphing a parabola requires precision and technique to achieve accuracy. Here's a simple guide to help you sketch the graph effectively:- **Start with the vertex**: It's the pivot of the parabola, which helps anchor the plot.- **Identify the y-intercept**: Substitute \( x = 0 \) into the function to find where the graph crosses the y-axis. For our function, this is the origin, or \( (0, 0) \).- **Select two additional points**: Choose convenient x-values to substitute into the equation for extra guidance on the graph's shape. For instance, using \( x = 1 \) and \( x = -1 \) gave us points \( (1, -4) \) and \( (-1, -2) \) respectively.- **Draw the parabola**: Connect these points smoothly, ensuring that the curve remains symmetrical around the vertex.Visualization transforms these calculated points into a meaningful graph, offering a complete picture of the quadratic equation in action.

Mathematical Verification

Verifying your graph ensures that your hand-drawn parabola aligns with the actual equation. This step confirms both the accuracy of your calculations and the precision of your drawing. Here's how to verify your graph:- **Use a graphing calculator**: Input your equation \( y = -3x^2 - x \) into the calculator. This confirms the structure and orientation of your graph through a quick visual.- **Cross-check plotted points**: Ensure the key points such as the vertex, y-intercept, and additional points align with those calculated.- **Double-check symmetry**: A downward-opening parabola like \( y = -3x^2 - x \), due to its negative \( a \) value, should exhibit symmetry about the line \( x = h \).Verification not only boosts confidence in your graphs but also sharpens analytical skills. With correct calculations and tools, you achieve mastery over graphing equations.