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Chapter 4: Problem 96

Find the distance from the vertex of the parabola \(f(x)=2(x-3)^{2}+5\) to the center of the circle \((x+3)^{2}+(y-1)^{2}=4\)

Short Answer

Expert verified
The distance is \(2\sqrt{13}\).

Step by step solution

01

Identify the vertex of the parabola

The given parabola is in the vertex form: \(f(x) = 2(x-3)^2 + 5\). The vertex form of a parabola is \(f(x) = a(x-h)^2 + k\), where (h,k) is the vertex. Here, the vertex is at \((3, 5)\).
02

Identify the center of the circle

The given circle equation is \((x+3)^2 + (y-1)^2 = 4\). The standard form of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where (h, k) is the center. So, the center of the circle is at \((-3, 1)\).
03

Use the distance formula

To find the distance between the vertex \((3, 5)\) and the center of the circle \((-3, 1)\), use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the points \((x_1, y_1) = (3, 5)\) and \((x_2, y_2) = (-3, 1)\), we get: \[ d = \sqrt{(-3-3)^2+(1-5)^2} = \sqrt{(-6)^2+(-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]

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Key Concepts

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

Vertex Form of a Parabola
The vertex form of a parabola is a useful way to describe the shape and position of a parabola. The general formula is \( f(x) = a(x-h)^2 + k \), where the point \((h, k)\) is the vertex of the parabola. This form makes it easy to see the key attributes:

Key Attributes

  • If \(a > 0\), the parabola opens upwards.
  • If \(a < 0\), it opens downwards.
  • The vertex \((h,k)\) is the highest or lowest point on the graph.
Understanding the vertex form helps in quickly graphing the parabola and finding important points. In the given exercise, the parabola \( f(x) = 2(x-3)^2 + 5 \) shows us that the vertex is at \((3, 5)\). This information is crucial for solving any problem involving the parabola.
Standard Form of a Circle
The standard form of a circle's equation helps easily identify its center and radius. The general form is \( (x-h)^2 + (y-k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius. This makes it straightforward to graph the circle.

Key Attributes

  • The point \((h, k)\) is the center of the circle.
  • The radius is \( r \), which is the distance from the center to any point on the circle.
In our exercise, the circle equation \((x+3)^2+(y-1)^2=4\) tells us the center is at \((-3, 1)\) and the radius is \(2\). Knowing the center allows us to find distances from this point.
Distance Formula
The distance formula is a key tool in geometry to find the distance between two points in a plane. The formula is \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. Here's a step-by-step guide on how to use it:

Steps to Use the Distance Formula

  • Identify the coordinates of the two points.
  • Subtract the x-coordinates and y-coordinates: \( (x_2 - x_1) \) and \( (y_2 - y_1) \).
  • Square the results: \( (x_2 - x_1)^2 \) and \( (y_2 - y_1)^2 \).
  • Sum the squares: \( (x_2 - x_1)^2 + (y_2 - y_1)^2 \).
  • Take the square root of the sum.
In the exercise, we needed to find the distance between the vertex \((3, 5)\) and the center of the circle \((-3, 1)\). Plugging these points into the formula gives us \[ d = \sqrt{(-3-3)^2+(1-5)^2} = \sqrt{(-6)^2+(-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]\.

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Most popular questions from this chapter

Complete the square for each quadratic function. $$ f(x)=x^{2}-10 x+7 $$

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=3 x^{2}-8 x+2\)

Under what circumstances is a linear function \(f(x)=m x+b\) odd? Can a linear function ever be even?

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In the United States, the birth rate \(B\) of unmarried women (births per 1000 unmarried women) for women whose age is \(a\) is modeled by the function \(B(a)=-0.33 a^{2}+19.17 a-213.37\) (a) What is the age of unmarried women with the highest birth rate? (b) What is the highest birth rate of unmarried women? (c) Evaluate and interpret \(B(40)\)
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