Problem 27 - Brief Applied Calculus

Question

Graph each function "by hand." [Note: Even if you have a graphing calculator, it is important to be able to sketch simple curves by finding a few important points.] $$ f(x)=2 x^{2}+4 x-16 $$

Step-by-Step Solution

Answer

Graph the parabola with the vertex at (-1, -18) using the points (-2, -16), (0, -16), and (1, -10).

Step 1: Identify the Function Type

The given function is a quadratic function of the form \( f(x) = ax^2 + bx + c \), where \( a = 2 \), \( b = 4 \), and \( c = -16 \). Quadratic functions graph as parabolas.

Step 2: Determine the Vertex

The vertex of the parabola can be found using the formula \( x = \frac{-b}{2a} \). Substituting the values, we get \( x = \frac{-4}{2 \cdot 2} = -1 \). Substitute \(-1\) into the function to find \( y \): \( f(-1) = 2(-1)^2 + 4(-1) - 16 = -18 \). The vertex is \((-1, -18)\).

Step 3: Find the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex. For this function, it is \( x = -1 \).

Step 4: Calculate Additional Points

Select additional \( x \)-values to help sketch the graph. Let's choose \( x = 0 \), \( x = -2 \), and \( x = 1 \).\- For \( x = 0 \): \( f(0) = 2(0)^2 + 4(0) - 16 = -16 \).\- For \( x = -2 \): \( f(-2) = 2(-2)^2 + 4(-2) - 16 = -16 \).\- For \( x = 1 \): \( f(1) = 2(1)^2 + 4(1) - 16 = -10 \).\These give the points \((0, -16)\), \((-2, -16)\), and \((1, -10)\).

Step 5: Sketch the Parabola

Use the vertex \((-1, -18)\), the calculated points \((0, -16)\), \((-2, -16)\), and \((1, -10)\), and the axis of symmetry \( x = -1 \) to sketch the parabola. Ensure it is a symmetric curve opening upwards since \( a = 2 \) is positive.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key point that defines the shape and location of the parabola on a graph. Understanding how to find the vertex will help in sketching the graph accurately without relying on technology.

For any quadratic function in the form \[ f(x) = ax^2 + bx + c \],learn how to find the vertex using the formula for the x-coordinate:

• The formula is: \[ x = \frac{-b}{2a} \]. By substituting the values of \( a \) and \( b \) from the function, you can find the x-coordinate of the vertex.

Once you determine \( x \), plug it back into the function to calculate the y-coordinate. This gives you the vertex as a point \((x, y)\).

For example, in the quadratic function \( f(x) = 2x^2 + 4x - 16 \), we find:

• \( x = \frac{-4}{2 \times 2} = -1 \)
• \( f(-1) = 2(-1)^2 + 4(-1) - 16 = -18 \)

Therefore, the vertex is \((-1, -18)\), which serves as the lowest point of the parabola since the parabola opens upwards.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two identical halves. This line passes through the vertex, ensuring that the parabola is symmetrical, meaning it looks the same on both sides.

To identify the axis of symmetry in a quadratic function, remember that it will always correspond to the x-coordinate of the vertex. This rule is fundamental as it aids in sketching the graph.

In the function \( f(x) = 2x^2 + 4x - 16 \), the vertex is \((-1, -18)\), so the axis of symmetry is:

• \( x = -1 \)

This axis acts like a mirror line,guiding us when plotting additional points and ensuring the graph correctly reflects on both sides of this line. Therefore, understanding and drawing the axis of symmetry is crucial in producing an accurate hand-drawn graph of a parabola.

Sketching Graphs by Hand

Sketching graphs by hand might seem challenging, but it is a valuable skill that helps deepen understanding of quadratic functions, without relying on graphing tools.

Here's a simple guide to hand-drawing a parabola from a quadratic equation:

• First, identify the vertex and axis of symmetry.
• Next, choose a few points on either side of the vertex to ensure accuracy. You might start with 3 or 4 points that include the vertex itself, such as \((x = 0)\), which often is simple to calculate.
• Plug the chosen x-values into the function to find corresponding y-values.
• Plot the vertex, axis of symmetry, and selected points on a coordinate plane.
• Finally, connect the points smoothly in a u-shape, ensuring symmetry around the axis.

For the function \( f(x) = 2x^2 + 4x - 16 \), you can use the points \((0, -16), (-2, -16), (1, -10)\), alongside the vertex \((-1, -18)\).Practicing these steps makes sketching graphs easier and more intuitive. Remember, the sketch does not have to be perfect, but it should clearly convey the main shape and characteristics of the parabola.