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Chapter 13: Problem 40

Write each equation of a parabola in standard form and graph it. Give the coordinates of the vertex. $$ y=x^{2}+4 x+5 $$

Short Answer

Expert verified
The vertex of the parabola is \((-2, 1)\). The equation in standard form is \(y = (x+2)^2 + 1\).

Step by step solution

01

Rewrite in Standard Form

The equation of a parabola is given as \( y = x^2 + 4x + 5 \). In order to rewrite it in the standard form \( y = a(x-h)^2 + k \), we will need to complete the square for the expression involving \( x \).
02

Complete the Square

To complete the square, we take the coefficient of \( x \), which is 4, divide it by 2, and then square it. This gives \( (\frac{4}{2})^2 = 4 \). Rewrite the equation by adding and subtracting 4 inside the equation: \( y = (x^2 + 4x + 4) + 5 - 4 \).
03

Factor Perfect Square

Notice that \( x^2 + 4x + 4 \) is a perfect square trinomial. It can be factored into \( (x+2)^2 \). Therefore, the equation becomes \( y = (x+2)^2 + 1 \).
04

Identify Vertex

Now that the equation is in the form \( y = (x-h)^2 + k \), we can identify the vertex \( (h,k) \). Here, \( h = -2 \) and \( k = 1 \), so the vertex of the parabola is \( (-2, 1) \).
05

Graph the Parabola

Plot the vertex \( (-2, 1) \) on the graph. The parabola opens upwards (since the coefficient of \( (x+2)^2 \) is positive). Use additional points obtained from the graph or calculation to complete the parabola's shape, such as points by substituting values around \( x = -2 \).

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Key Concepts

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

Parabola
A parabola is a U-shaped curve that you often encounter in mathematics, particularly when dealing with quadratic equations. It's a graph that represents a quadratic function, which is any function that can be written as: \[ y = ax^2 + bx + c \]In this form, the parabola can open upwards or downwards, dictated by the coefficient \(a\).
Key characteristics of a parabola:
  • The vertex can be either a maximum or a minimum point, depending on the direction the parabola opens.
  • If \(a > 0\), the parabola opens upwards, and the vertex is the lowest point.
  • If \(a < 0\), it opens downwards, and the vertex is the highest point.
  • The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
Understanding parabolas is crucial because they appear in various contexts, from physics to engineering, whenever you deal with quadratic relationships. Parabolas can model real-world phenomena like projectile motion or antenna signal focusing.
Vertex
The vertex of a parabola is a significant point that represents its peak or trough, depending on the orientation. For the quadratic equation \(y = ax^2 + bx + c\), the vertex can be found using a specific transformation. The standard form of a quadratic equation 鈥� \(y = a(x-h)^2 + k\) 鈥� highlights this vertex readily.In this standard form:
  • \(h\) is the x-coordinate of the vertex.
  • \(k\) is the y-coordinate of the vertex.
  • Thus, the vertex is at \((h, k)\).
In our exercise, by completing the square, we transformed \(y = x^2 + 4x + 5\) into \(y = (x+2)^2 + 1\). From here, we found the vertex: \((-2, 1)\).
The vertex is not just a point; it helps determine many properties of a parabola:
  • I establishes the line of symmetry.
  • Assists in identifying the direction in which the parabola opens.
  • It helps solve optimization problems in various fields, as it represents maximum or minimum values.
Standard Form
The standard form of a quadratic equation is quite useful in understanding the nature of parabolas. The standard form is expressed as: \[ y = a(x - h)^2 + k \]This format directly reveals the vertex and conveys the shape of the parabola.
Here's why the standard form is valuable:
  • It directly gives the vertex \((h, k)\), simplifying the process of graphing.
  • It makes it easier to determine how the parabola shifts and reflects.
  • Using \(a\), we can immediately know if the parabola opens upwards or downwards and gauge its width.
To rewrite a quadratic equation like \(y = x^2 + 4x + 5\) into the standard form, we completed the square. This gave us \(y = (x+2)^2 + 1\), making it easier to see that the vertex is at \((-2, 1)\). This transformation helps in graphing and analyzing the properties of the parabola clearly, offering insights easier and faster than using the general quadratic form.

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Most popular questions from this chapter

Solve each system of equations for real values of x and y. $$ \left\\{\begin{array}{l} x^{2}-y=0 \\ x^{2}-4 x+y=0 \end{array}\right. $$

Write each equation in standard form, if it is not alreacty so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ y=-4(x-4)^{2}-4 $$

Investing. Grant receives \(\$ 225\) annual income from one investment. Jeff invested \(\$ 500\) more than Grant, but at an annual rate of \(1 \%\) less. Jeff's annual income is \(\$ 240 .\) What are the amount and rate of Grant's investment?

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Find the center and radius of each circle. A. \((x-4)^{2}+(y+7)^{2}=28\) B. \((x+4)^{2}+(y-7)^{2}=28\)
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