<< Chapter < Page Chapter >> Page >

Finding the y -intercept by substituting x = 0 into the equation is easy, isn’t it? But we needed to use the Quadratic Formula to find the x -intercepts in [link] . We will use the Quadratic Formula again in the next example.

Graph y = 2 x 2 4 x 3 .

Solution

.
The equation y has one side.
Since a is 2, the parabola opens upward.
.
To find the axis of symmetry, find x = b 2 a . .
.
.
The axis of symmetry is x = 1 .
The vertex on the line x = 1 . .
Find y when x = 1 . .
.
.
The vertex is ( 1 , 5 ) .
The y -intercept occurs when x = 0 . .
Substitute x = 0 . .
Simplify. .
The y- intercept is ( 0 , −3 ) .
The point ( 0 , −3 ) is one unit to the left of the line of symmetry.
The point one unit to the right of the line of symmetry is ( 2 , −3 )
Point symmetric to the y- intercept is ( 2 , −3 ) .
The x -intercept occurs when y = 0 . .
Substitute y = 0 . .
Use the Quadratic Formula. .
Substitute in the values of a, b, c. .
Simplify. .
Simplify inside the radical. .
Simplify the radical. .
Factor the GCF. .
Remove common factors. .
Write as two equations. .
Approximate the values. .
The approximate values of the x- intercepts are ( 2.5 , 0 ) and ( −0.6 , 0 ) .
Graph the parabola using the points found. .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Graph the parabola y = 5 x 2 + 10 x + 3 .

y : ( 0 , 3 ) ; x : ( −1.6 , 0 ) , ( −0.4 , 0 ) ;
axis: x = −1 ; vertex: ( −1 , −2 ) ;
The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 5. The vertex is at the point (-1,-2). Three other points are plotted on the curve at (0, 3), (-1.6, 0), (-0.4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.

Got questions? Get instant answers now!

Graph the parabola y = −3 x 2 6 x + 5 .

y : ( 0 , 5 ) ; x : ( 0.6 , 0 ) , ( −2.6 , 0 ) ;
axis: x = −1 ; vertex: ( −1 , 8 ) ;
The graph shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (-1, 8). Three other points are plotted on the curve at (0, 5), (0.6, 0) and (-2.6, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.

Got questions? Get instant answers now!

Solve maximum and minimum applications

Knowing that the vertex    of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic equation. The y -coordinate of the vertex is the minimum y -value of a parabola that opens upward. It is the maximum y -value of a parabola that opens downward. See [link] .

This figure shows two graphs side by side. The left graph shows an downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the upper right quadrant. The vertex is labeled “maximum”. The right graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the lower right quadrant. The vertex is labeled “minimum”.

Minimum or maximum values of a quadratic equation

The y -coordinate of the vertex of the graph of a quadratic equation is the

  • minimum value of the quadratic equation if the parabola opens upward.
  • maximum value of the quadratic equation if the parabola opens downward.

Find the minimum value of the quadratic equation y = x 2 + 2 x 8 .

Solution

.
Since a is positive, the parabola opens upward.
The quadratic equation has a minimum.
Find the axis of symmetry. .
.
.
The axis of symmetry is x = −1 .
The vertex is on the line x = −1 . .
Find y when x = −1 . .
.
.
The vertex is ( −1 , −9 ) .
Since the parabola has a minimum, the y- coordinate of the vertex is the minimum y- value of the quadratic equation.
The minimum value of the quadratic is −9 and it occurs when x = −1 .
Show the graph to verify the result. .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the maximum or minimum value of the quadratic equation y = x 2 8 x + 12 .

The minimum value is −4 when x = 4 .

Got questions? Get instant answers now!

Find the maximum or minimum value of the quadratic equation y = −4 x 2 + 16 x 11 .

The maximum value is 5 when x = 2 .

Got questions? Get instant answers now!

We have used the formula

h = −16 t 2 + v 0 t + h 0

to calculate the height in feet, h , of an object shot upwards into the air with initial velocity, v 0 , after t seconds.

This formula is a quadratic equation in the variable t , so its graph is a parabola. By solving for the coordinates of the vertex, we can find how long it will take the object to reach its maximum height. Then, we can calculate the maximum height.

The quadratic equation h = −16 t 2 + v 0 t + h 0 models the height of a volleyball hit straight upwards with velocity 176 feet per second from a height of 4 feet.

  1. How many seconds will it take the volleyball to reach its maximum height?
  2. Find the maximum height of the volleyball.

Solution

h = −16 t 2 + 176 t + 4

Since a is negative, the parabola opens downward.

The quadratic equation has a maximum.


  1. Find the axis of symmetry. t = b 2 a t = 176 2 ( −16 ) t = 5.5 The axis of symmetry is t = 5.5 . The vertex is on the line t = 5.5 . The maximum occurs when t = 5.5 seconds.

  2. Find h when t = 5.5 . .
    .
    Use a calculator to simplify. .
    The vertex is ( 5.5 , 488 ) .
    Since the parabola has a maximum, the h- coordinate of the vertex is the maximum y -value of the quadratic equation. The maximum value of the quadratic is 488 feet and it occurs when t = 5.5 seconds.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
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
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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
Practice Key Terms 6

Get the best Elementary algebra course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask