/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Find an equation for the parabol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 18

Find an equation for the parabola that satisfies the given conditions. Vertex \((5,-3) ;\) axis parallel to the \(y\) -axis; passes through \((9,5) .\)

Short Answer

Expert verified
The equation of the parabola is \(y = \frac{1}{2}(x - 5)^2 - 3\).

Step by step solution

01

Identify the Standard Form of a Parabola

Since the parabola has a vertex at \((5, -3)\) and the axis is parallel to the \(y\)-axis, its equation is of the form \(y = a(x - h)^2 + k\). Here, \((h, k)\) is the vertex, so \(h = 5\) and \(k = -3\). Thus, the equation becomes \(y = a(x - 5)^2 - 3\).
02

Substitute the Point in the Equation

The parabola passes through the point \((9, 5)\). Substitute \(x = 9\) and \(y = 5\) into the equation \(y = a(x - 5)^2 - 3\). This gives us \(5 = a(9 - 5)^2 - 3\).
03

Solve for a

Simplify the equation \(5 = a(4)^2 - 3\) to find the value of \(a\). This becomes \(5 = 16a - 3\).\ Re-arrange to find \(16a = 8\), which gives \(a = \frac{1}{2}\).
04

Write the Final Equation of the Parabola

Substitute \(a = \frac{1}{2}\) back into the equation from Step 1. The final equation becomes \(y = \frac{1}{2}(x - 5)^2 - 3\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Vertex Form
The vertex form of a parabola is particularly useful because it provides a clear picture of the parabola's vertex. The formula for the vertex form is given as \( y = a(x - h)^2 + k \). Here, \((h, k)\) represents the coordinates of the vertex of the parabola.
The parameter \(a\) indicates the parabola's vertical stretch and direction. When \(a\) is positive, the parabola opens upwards, while a negative \(a\) means it opens downwards.
  • Vertex: The point \((h, k)\) is the vertex, which represents the maximum or minimum point of the parabola, depending on its direction.
  • Transformations: The values of \(h\) and \(k\) shift the parabola horizontally and vertically, respectively.
  • Example: If the vertex is \((5, -3)\), the vertex form is \( y = a(x - 5)^2 - 3 \).
To find \(a\), substitute a point on the parabola into the equation. Solve for \(a\) using these known values.
Y-axis Symmetry
Y-axis symmetry in a parabola implies that its graph is mirrored across the vertical line, or the axis of symmetry. For parabolas that open upwards or downwards, this axis of symmetry goes through the vertex's x-coordinate, which aligns parallel to the y-axis.
A parabola with its axis of symmetry parallel to the y-axis will have equations that maintain this symmetry. This happens because the square term \((x - h)^2\) ensures that each side of the parabola is symmetric in respect to its axis
  • Equation Consistency: The vertex form \(y = a(x - h)^2 + k\) sustains this symmetry, with all x-values equidistant from \(h\), resulting in identical y-values.
  • Graphical Representation: You can draw a line through the vertex parallel to the y-axis to visualize this symmetry.
Such a symmetry is a fundamental characteristic of parabola graphs, enabling easier sketching and understanding of their shape.
Standard Form of a Parabola
The standard form of a parabola is another popular method of expressing the quadratic equation. It is typically written as:
\[ y = ax^2 + bx + c \]Unlike the vertex form, the standard form gives direct coefficients of the quadratic equation, focusing more on the parabola's general shape and intercepts.
  • Direct Calculation: While the vertex form provides easy insights into the vertex, the standard form is more focused on comprehensive calculation methods like using the quadratic formula.
  • Conversion from Vertex Form: To convert an equation from vertex to standard form, expand the square in the vertex form and simplify.
  • Intercepts: The constant term \(c\) often indicates the y-intercept of the parabola.
Understanding both forms allows for flexibility in solving and graphing quadratic equations, as they offer different insights into the parabola's properties.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The sphere of radius \(a\) generated by revolving the semicircle \(r=a\) in the upper half-plane about the polar axis.

If \(f^{\prime}(t)\) and \(g^{\prime}(t)\) are continuous functions, and if no segment of the curve $$ x=f(t), \quad y=g(t) \quad(a \leq t \leq b) $$ is traced more than once, then it can be shown that the area of the surface generated by revolving this curve about the \(x\) -axis is $$ S=\int_{a}^{b} 2 \pi y \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t $$ and the area of the surface generated by revolving the curve about the \(y\) -axis is $$ S=\int_{a}^{b} 2 \pi x \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t $$ [The derivations are similar to those used to obtain Formulas (4) and (5) in Section 6.5.] Use the formulas above in these exercises. Find the area of the surface generated by revolving \(x=t^{2}\), \(y=3 t(0 \leq t \leq 2)\) about the \(x\) -axis.

Find the area inside the curve \(r^{2}=\sin 2 \theta\).

Find a polar equation for the conic that has its focus at the pole and satisfies the stated conditions. Points are in polar coordinates and directrices in rectangular coordinates for simplicity. (In some cases there may be more than one conic that satisfies the conditions.) (a) Ellipse; \(e=\frac{3}{4} ;\) directrix \(x=2\). (b) Parabola; directrix \(x=1\). (c) Hyperbola; \(e=\frac{4}{3}\); directrix \(y=3\).

True-False Determine whether the statement is true or false. Explain your answer. The polar coordinate pairs \((-1, \pi / 3)\) and \((1,-2 \pi / 3)\) describe the same point.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.