Question

You are on a research boat in the ocean. You see a penguin jump out of the water. The path followed by the penguin is given by $$ h=-0.05 x^{2}+1.178 x $$ where \(h\) is the height (in feet) the penguin jumps out of the water and \(x\) is the horizontal distance (in feet) traveled by the penguin over the water. Sketch a graph of the equation.

Short answer

The graph is a downward parabola with vertex calculated as per the formula, it's symmetry axis a vertical line through the vertex, opening downwards due to the negative coefficient of the quadratic term.

Step-by-step solution

Identify The Quadratic Function

Recognize the function \(h=-0.05 x^{2}+1.178 x\) as a quadratic function, in the form of \(y=ax^2+bx+c\), where \(a=-0.05\), \(b=1.178\), and \(c=0\) because there is no constant term.

Identify the Vertex

The vertex of a parabola \(y=ax^2+bx+c\) is given by \((-b/2a, f(-b/2a))\). Here, calculate \(x=-b/2a = -1.178/(-0.1)\) to get the x-coordinate of the vertex. Then substitute this into the equation to get the y-coordinate.

Draw the Axis of Symmetry

Draw a vertical line through the x-coordinate of the vertex. This is called the axis of symmetry of the parabola.

Identify the Direction of the Parabola

Since the coefficient \(a\), which is -0.05, is negative, the parabola will open downwards.

Sketch the Parabola

With the vertex and direction known, you can now sketch the graph of the parabola. It will be a downward-opening curve that is symmetric about the axis of symmetry.

Key concepts

Vertex of a Parabola

In the context of quadratic functions, the vertex is the highest or lowest point on the graph, known as a parabola. The vertex is a cornerstone concept as it serves not only as a key point to sketch the graph but also gives insights into the function's maximum or minimum values.

For a quadratic function in the form of \(y = ax^2 + bx + c\), the vertex can be calculated using the formula \((h, k) = (-b/2a, f(-b/2a))\), where \(h\) and \(k\) are the x and y coordinates of the vertex, respectively. In our exercise, we have the quadratic function \(h=-0.05x^2+1.178x\). Using the formula, we calculate the x-coordinate of the vertex as \(-b/2a\), which gives us \(-1.178/(-0.1)\). Substituting this value back into the function \(h\) provides us with the y-coordinate of the vertex.

• The vertex is crucial for determining the direction of the opening of the parabola (whether it faces up or down).
• It helps in understanding the maximum height or the maximum distance covered.
• In our example, it reveals the highest point of the penguin's jump.

Axis of Symmetry

The axis of symmetry is a vertical line that splits the parabola into two mirror images. It passes through the vertex, and its equation is \(x = h\) where \(h\) is the x-coordinate of the vertex. For students and others working with quadratic functions, the axis of symmetry is vital as it simplifies graphing and analyzing the parabola's properties.

In the solution to our exercise, we drew a vertical line through the x-coordinate of the vertex obtained, which represents the axis of symmetry. If you were to fold the graph along this line, both sides would align perfectly, demonstrating the symmetry.

• The axis of symmetry can be used to predict the parabola's behavior on one side if the behavior on the other side is known.
• It is an essential reference for solving problems related to quadratic functions in real-world scenarios, such as determining the path of a projectile.

Graphing Parabolas

The process of graphing parabolas involves plotting a curve represented by a quadratic function on a coordinate plane. When graphing, we look for several components: the vertex, the axis of symmetry, the direction of opening (upwards or downwards), and the width or steepness of the parabola.

To graph our function \(h=-0.05x^2+1.178x\), we start by plotting the vertex. Next, we draw the axis of symmetry passing through the vertex. Since the coefficient of \(x^2\) is negative, the parabola opens downward, which is a critical observation for sketching the shape correctly.

• The procedure of graphing helps visualize the trajectory of an object, such as our penguin's jump.
• Students can utilize this visualization to understand the maximum and minimum points of real-life problems represented by quadratic functions.

Remembering to plot points symmetrically on either side of the axis of symmetry can make graphing smoother and more accurate.