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Chapter 10: Problem 39

Write an equation for each parabola with vertex at the origin. through \((3,2),\) symmetric with respect to the \(x\) -axis

Short Answer

Expert verified
The equation of the parabola is \(y = \frac{2}{9}x^2\).

Step by step solution

01

Determine the general form of a parabola with vertex at the origin

A parabola with its vertex at the origin can take the form of either \(y = ax^2\) or \(x = ay^2\) depending on its orientation. Since it is symmetric with respect to the x-axis, we'll use the form \(y = ax^2\).
02

Substitute the given point into the equation

The point \((3, 2)\) is on the parabola. Substitute \(x = 3\) and \(y = 2\) into the equation \(y = ax^2\). This gives us:\(2 = a (3)^2\).
03

Solve for the coefficient a

Solve the equation \(2 = a (3)^2\) to find the value of \(a\).\(2 = 9a\) \(a = \frac{2}{9}\)
04

Write the final equation of the parabola

Now that we have \(a\), we can write the equation of the parabola as: \(y = \frac{2}{9}x^2\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertex at the origin
A parabola with its vertex at the origin means that the highest or lowest point of the parabola is at \(0,0\). The vertex is central to the shape and direction of the parabola.
When the vertex is at \(0,0\), the standard forms of the equation for a parabola are \(y = ax^2\) and \(x = ay^2\).
These forms depend on the orientation of the parabola. If the parabola opens upwards or downwards, it's \(y = ax^2\). If it opens to the right or left, it's \(x = ay^2\).
Knowing the vertex is at the origin simplifies the equation because there is no need for \(h\) and \(k\) terms in the vertex form \(y = a(x-h)^2 + k\).
Thus, understanding the vertex position helps to directly use the standard form and move forward with easily substituting values.
symmetry with respect to x-axis
Symmetry with respect to the x-axis means that the parabola is mirrored along the x-axis.
If you fold the graph along the x-axis, both halves will overlap exactly.
For a parabola symmetric with respect to the x-axis, you will use the equation \(y = ax^2\). This is because the symmetry tells us the parabola opens either upwards or downwards.
In such cases, given any point \(x,y\) on the curve, the point \(-x,y\) will also lie on the curve.
Understanding symmetry helps ensure the correct standard equation format is used and any calculations made respect this balance.
coefficients in quadratic equations
In the quadratic equation \(y = ax^2\), the coefficient \(a\) determines the width and direction of the parabola.
A positive \(a\) makes the parabola open upwards, and a negative \(a\) makes it open downwards.
The value of \(a\) affects the 'stretch' or 'compression' of the parabola. A larger absolute value of \(a\) results in a narrower parabola, while a smaller absolute value of \(a\) results in a wider one.
To find the specific value of \(a\), you can substitute a known point that lies on the parabola into the general equation and solve for \(a\).
For example, if through the point \((3,2)\), setting up the equation \(2 = a(3)^2\) and solving for \(a\) gives \(a = \frac{2}{9}\).
Understanding coefficients is crucial because it tells us exactly how the parabola will behave in the Cartesian plane.

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Most popular questions from this chapter

Find the eccentricity of each hyperbola to the nearest tenth. $$\frac{x^{2}}{2}-\frac{y^{2}}{18}=1$$

Solve each problem. When an object moves under the influence of a constant force (without air resistance), its path is parabolic. This would occur if a ball were thrown near the surface of a planet or other celestial object. Suppose two balls are simultaneously thrown upward at a \(45^{\circ}\) angle on two different planets. If their initial velocities are both \(30 \mathrm{mph}\), then their \(x y\) -coordinates in feet at time \(x\) in seconds can be modeled by the following equation. $$ y=x-\frac{g}{1922} x^{2} $$ Here \(g\) is the acceleration due to gravity. The value of \(g\) will vary depending on the mass and size of the planet. (Source: Zeilik, M., and S. Gregory, Introductory Astronomy and Astrophysics, Fourth Edition, Brooks/Cole.) (a) For Earth \(g=32.2,\) while for Mars \(g=12.6 .\) Find the two equations, and graph on the same screen of a graphing calculator the paths of the two balls thrown on Earth and Mars. Use the window \([0,180]\) by \([0,120] .\) (Hint: If possible, set the mode on your graphing calculator to simultaneous.) (b) Determine the difference in the horizontal distances traveled by the two balls.

$$\text {Give the focus, directrix, and axis of symmetry for each parabola.}$$ $$y=-\frac{1}{9} x^{2}$$

Write an equation for each hyperbola. $$y \text { -intercepts } \pm 12 ; \text { foci at }(0,-15),(0,15)$$

Solve each problem. The coordinates in miles for the orbit of the artificial satellite Explorer VII can be modeled by the equation $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$$ where \(a=4465\) and \(b=4462 .\) Earth's center is located at one focus of the elliptical orbit. (Source: Loh, W., Dynamics and Thermodynamics of Planetary Entry, Prentice-Hall; Thomson, W., Introduction to Space Dynamics, John Wiley and Sons.) (a) Graph both the orbit of Explorer VII and the Earth's surface on the same coordinate axes if the average radius of Earth is 3960 mi. Use the window \([-6750,6750]\) by \([-4500,4500].\) (b) Find the maximum and minimum heights of the satellite above Earth's surface.
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