18.) Graph the circle and line to decide if the line intersects and if so if it is a secant of tangent.. 3 y-1 = -(x+4) (x + 2)2 + (y - 2)2 = 9 %3D -3 -2. -5-4-3-2 -1 1 2 3 4 5 -2 -4 -5 543 2-
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![### Graphing Circles and Lines to Determine Intersection
**Objective:**
To graph a given circle and a line, then determine if they intersect. If they do intersect, also determine whether the line is a secant or tangent to the circle.
**Problem Statement:**
Graph the circle and line based on the following equations and decide if the line intersects and if so, determine if it is a secant or a tangent.
**Equations:**
1. Line: \(y - 1 = -\frac{3}{2}(x + 4)\)
2. Circle: \((x + 2)^2 + (y - 2)^2 = 9\)
**Steps to Solve the Problem:**
1. **Rewrite the Equations:**
- **Line Equation:**
\[ y - 1 = -\frac{3}{2}(x + 4) \]
Solving for y:
\[ y = -\frac{3}{2}(x + 4) + 1 \]
\[ y = -\frac{3}{2}x - 6 + 1 \]
\[ y = -\frac{3}{2}x - 5 \]
- **Circle Equation:**
\[ (x + 2)^2 + (y - 2)^2 = 9 \]
This represents a circle with center at \((-2, 2)\) and radius \(3\).
2. **Graphing the Equations:**
- **Circle:**
- The center of the circle is \((-2, 2)\).
- The radius is \(3\), hence the points at a distance of 3 units from \((-2, 2)\) are along the circumference of the circle.
- **Line:**
- The line equation \(y = -\frac{3}{2}x - 5\) will intersect the y-axis at \((0, -5)\).
- The slope of the line is \(-\frac{3}{2}\); for every 2 units increase in x, y decreases by 3 units.
3. **Graph in Detail:**
- The graph has an x-axis and a y-axis marked from \(-5\) to \(5\) with grid lines.
- Plot the center of the circle at \((-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0222f4aa-a8ff-4b8a-92fa-f3e9206b9deb%2F98d7f3ec-a45f-41fc-84a1-c89e1934feac%2F5cu8wp_processed.jpeg&w=3840&q=75)
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