# 10.5 Graphing quadratic equations  (Page 4/15)

 Page 4 / 15

Find the intercepts of the parabola $y=9{x}^{2}+12x+4.$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,4\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{2}{3},0\right)$

## Graph quadratic equations in two variables

Now, we have all the pieces we need in order to graph a quadratic equation in two variables. We just need to put them together. In the next example, we will see how to do this.

## How to graph a quadratic equation in two variables

Graph $y={x}^{2}-6x+8$ .

## Solution

Graph the parabola $y={x}^{2}+2x-8.$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,-8\right)$ ; $x\text{:}\phantom{\rule{0.2em}{0ex}}\left(2,0\right),\left(-4,0\right)$ ;
axis: $x=-1$ ; vertex: $\left(-1,-9\right)$ ;

Graph the parabola $y={x}^{2}-8x+12.$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,12\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(2,0\right),\left(6,0\right);$
axis: $x=4;\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}\left(4,-4\right)$ ;

## Graph a quadratic equation in two variables.

1. Write the quadratic equation with $y$ on one side.
2. Determine whether the parabola opens upward or downward.
3. Find the axis of symmetry.
4. Find the vertex.
5. Find the y -intercept. Find the point symmetric to the y -intercept across the axis of symmetry.
6. Find the x -intercepts.
7. Graph the parabola.

We were able to find the x -intercepts in the last example by factoring. We find the x -intercepts in the next example by factoring, too.

Graph $y=\text{−}{x}^{2}+6x-9$ .

## Solution

 The equation y has on one side. Since a is $-1$ , the parabola opens downward. To find the axis of symmetry, find $x=-\frac{b}{2a}$ . The axis of symmetry is $x=3.$ The vertex is on the line $x=3.$ Find y when $x=3.$ The vertex is $\left(3,0\right).$ The y -intercept occurs when $x=0.$ Substitute $x=0.$ Simplify. The point $\left(0,-9\right)$ is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is $\left(6,-9\right).$ Point symmetric to the y- intercept is $\left(6,-9\right)$ The y -intercept is $\left(0,-9\right).$ The x -intercept occurs when $y=0.$ Substitute $y=0.$ Factor the GCF. Factor the trinomial. Solve for x . Connect the points to graph the parabola.

Graph the parabola $y=-3{x}^{2}+12x-12.$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,-12\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(2,0\right);$
axis: $x=2;\phantom{\rule{0.2em}{0ex}}\text{vertex:}\left(2,0\right)$ ;

Graph the parabola $y=25{x}^{2}+10x+1.$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,1\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{1}{5},0\right);$
axis: $x=-\frac{1}{5};\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}\left(-\frac{1}{5},0\right)$ ;

For the graph of $y=-{x}^{2}+6x-9$ , the vertex and the x -intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation $0=\text{−}{x}^{2}+6x-9$ is 0, so there is only one solution. That means there is only one x -intercept, and it is the vertex of the parabola.

How many x -intercepts would you expect to see on the graph of $y={x}^{2}+4x+5$ ?

Graph $y={x}^{2}+4x+5$ .

## Solution

 The equation has y on one side. Since a is 1, the parabola opens upward. To find the axis of symmetry, find $x=-\frac{b}{2a}.$ The axis of symmetry is $x=-2.$ The vertex is on the line $x=-2.$ Find y when $x=-2.$ The vertex is $\left(-2,1\right).$ The y -intercept occurs when $x=0.$ Substitute $x=0.$ Simplify. The point $\left(0,5\right)$ is two units to the right of the line of symmetry. The point two units to the left of the line of symmetry is $\left(-4,5\right).$ The y -intercept is $\left(0,5\right).$ Point symmetric to the y- intercept is $\left(-4,5\right)$ . The x - intercept occurs when $y=0.$ Substitute $y=0.$ Test the discriminant. ${b}^{2}-4ac$ ${4}^{2}-4\cdot 15$ $16-20$ $\phantom{\rule{1em}{0ex}}-4$ Since the value of the discriminant is negative, there is no solution and so no x- intercept. Connect the points to graph the parabola. You may want to choose two more points for greater accuracy.

Graph the parabola $y=2{x}^{2}-6x+5.$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,5\right);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none};$
axis: $x=\frac{3}{2};\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}\left(\frac{3}{2},\frac{1}{2}\right)$ ;

Graph the parabola $y=-2{x}^{2}-1.$

$y\text{:}\phantom{\rule{0.2em}{0ex}}\left(0,-1\right);x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none};$
axis: $x=0;\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}\left(0,-1\right)$ ;

what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes ashley Reply app is wrong how can 350 be divisible by 3. Raheem Reply June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold? Susanna Reply Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct? Georgie @Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple Ashley @Geogie my bad that was meant for u Ashley Hi everyone, I'm glad to be connected with you all. from France. Lorris Reply I'm getting "math processing error" on math problems. Anyone know why? Ray Reply Can you all help me I don't get any of this Jade Reply 4^×=9 Alberto Reply Did anyone else have trouble getting in quiz link for linear inequalities? Sireka Reply operation of trinomial Justin Reply y=2×+9 Jacob Reply Keshad gets paid$2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay? Hector Reply Mayra has$124 in her checking account. She writes a check for $152. What is the New Balance in her checking account? REVOLUTION Reply -28$
ashley
-\$28
Stephanie