Math 22 Graph of Equation

The Graph of an Equation

The graph of an equation is the set of all points that are solutions of the equation.

In this section, we use point-plotting method. With this method, you construct a table of values that consists of several solution points of the equation

For example, sketch the graph of $y=2x+1$. We can construct the table below by plugging points for $x$.

x	0	1	2	3
y=2x+1	1	3	5	7

So, we can sketch the graph from those order pairs.

Intercepts of a Graph

Some solution points have zero as either the $x$-coordinate or the $y$-coordinate. These points are called intercepts because they are the points at which the graph intersects the $x$- or $y$-axis.

 To find $x$-intercepts, let $y$ be zero and solve the equation for $x$.
 
 To find $y$-intercepts, let $x$ be zero and solve the equation for $y$.

Example Find the x-intercepts and y-intercepts of the function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=x^{2}-2x}

Solution:
x-intercept: Let $y=0$, so $x^{2}-2x=0$, hence $x(x-2)=0$, therefore, $x=0$ or $x=2$
y-intercept: Let $x=0$, so $y=(0)^{2}-2(0)=0$
Answer: $(0,0)$ and $(2,0)$ are x-intercepts
$(0,0)$ is y-intercept

Circles

 The standard form of the equation of a circle is
 
 $(x-h)^{2}+(y-k)^{2}=r^{2}$
 
 The point Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (h,k)}
 is the center of the circle, and the positive number Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r}
 is the radius of the circle

In general, to write an equation of a circle, we need to know radius Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r} and the center Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (h,k)} .

Example Given that the point Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (2,1)} is on the circle centered at (3,4). Find the equation of a circle.

Solution:
We need to know the radius and the center in order to write the equation. The center is given at $(3,4)$. It is left to find the radius.
Radius is the distance between the center and a point on the circle. So, radius $r$ is the distance between Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (2,1)} and $(3,4)$.
So, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r={\sqrt {(2-3)^{2}+(1-4)^{2}}}={\sqrt {1+9}}={\sqrt {10}}}
Now, write the equation of the circle with radius Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r={\sqrt {10}}} and center $(3,4)$ to get:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (x-3)^{2}+(y-4)^{2}=10}

Notes

Distance Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D} between $(x_{1},y_{1})$ and $(x_{2},y_{2})$ can be calculated by using Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}}

Points of Intersection

An ordered pair that is a solution of two different equations is called a point of intersection of the graphs of the two equations

For example, find the point(s) of intersection of two equations Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=2x-1} and $y=3x+4$.

The order pairs that satisfy both of these equation should have the same $y$ value, so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=2x-1=3x+4}

Then, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2x-1=3x+4}

Therefore Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=-5}

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