Problem 47 - Algebra and Trigonometry with Analytic Geometry

Question

Flights of leaping animals typically have parabolic paths. The figure on the next page illustrates a frog jump superimposed on a coordinate plane. The length of the leap is 9 feet, and the maximum height off the ground is 3 feet. Find a standard equation for the path of the frog.

Step-by-Step Solution

Answer

The equation for the frog's path is \( y = -\frac{4}{27}(x-4.5)^2 + 3 \).

Step 1: Determine the standard form of a parabolic equation

The standard form of a parabolic equation is given by: \( y = ax^2 + bx + c \). We know a parabola that opens downwards can also be expressed as: \( y = a(x-h)^2 + k \). Since the parabolic path is symmetric about the x-axis, we'll use this second form where \( h \) and \( k \) are the coordinates of the vertex.

Step 2: Identify the vertex

The frog reaches the maximum height of 3 feet at the midpoint of its jump. This means the vertex of the parabola is at \( (4.5, 3) \) since the leap length is 9 feet, making the midpoint \( 4.5 \) feet.

Step 3: Establish coordinates of other points

The leap starts and ends on the ground, which means it crosses the x-axis at the points \( (0,0) \) and \( (9,0) \). These are the points where the height \( y = 0 \).

Step 4: Write the equation in vertex form

Using vertex form \( y = a(x-h)^2 + k \), where \( h = 4.5 \) and \( k = 3 \), substitute to get: \( y = a(x-4.5)^2 + 3 \).

Step 5: Use a point to find \( a \)

Substitute one of the x-intercept points into the equation to find \( a \). Using \( (0,0) \), we have: \( 0 = a(0-4.5)^2 + 3 \). Simplifying gives: \( 0 = 20.25a + 3 \). Solving for \( a \), we find \( a = -\frac{3}{20.25} \), or approximately \( a = -\frac{4}{27} \).

Step 6: Write the completed equation

Substitute the value of \( a \) back into the equation: \( y = -\frac{4}{27}(x-4.5)^2 + 3 \). This is the standard equation that represents the path of the frog's jump.

Key Concepts

Vertex Form Equation

The vertex form of a quadratic parabola is a powerful way to express its equation. It is given by the formula \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

• The vertex represents the highest or lowest point of the parabola, depending on its orientation. For upward-opening parabolas, it is the minimum point, and for downward-opening parabolas, it is the maximum point.
• In the context of the frog’s leap, the vertex is the peak height the frog reaches during the jump, located symmetrically in the middle of the leap's path length.

Representing a parabola in vertex form simplifies the process of finding its maximum or minimum value because the vertex is explicitly stated in the equation. To find the coefficients \(a\), \(h\), and \(k\), you can use known points on the parabolic path such as the vertex and any other point, like intercepts or given coordinates.
When solving the exercise, identifying that the vertex is at \((4.5, 3)\) made it straightforward to substitute these values into the vertex form equation.

Symmetry of Parabolas

A key feature of parabolas is their symmetry. This symmetry is about a vertical line called the axis of symmetry, which passes through the vertex.

• For a parabola in vertex form, \( y = a(x-h)^2 + k \), the axis of symmetry is the vertical line \( x = h \). In our frog leap example, this vertical line is \( x = 4.5 \).
• This symmetry means that a parabola is a mirror image of itself about this axis. This property helps in determining corresponding points on the parabola's two sides.
• Because parabolas are symmetrical, any point \(x\) on the left side of the axis will have a corresponding point at \(2h - x\) with the same \(y\)-coordinate.

Understanding this symmetry is crucial when analyzing the motion of objects, like the frog, along a parabolic path. It simplifies calculations by confirming that the parabolic trajectory is equal on either side of the vertex.

X-Intercepts of Parabolas

The x-intercepts of a parabola are the points where the parabola crosses the x-axis. At these points, the value of \( y \) is zero. In the quadratic equation \( ax^2 + bx + c = 0 \) or in vertex form, the x-intercepts give insight into the roots of the equation.

• In vertex form, you can convert to standard form to find intercepts using the quadratic formula, or simply substitute \( y = 0 \) and solve for \( x \).
• For the frog’s jump, the parabola crosses the x-axis at \( x = 0 \) and \( x = 9 \). These are the points where the frog starts and ends its jump.
• To find the x-intercepts in our problem, we used these points to confirm the value of the coefficient \( a \) after identifying the vertex.

Understanding x-intercepts is vital for grasping the full scope of an object's motion through parabolic trajectories, as they define where the object leaves and re-joins the ground. These intercepts provide the boundaries within which the parabolic motion occurs.