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Access these online resources for additional instruction and practice with quadratic equations.

Key equations

general form of a quadratic function f ( x ) = a x 2 + b x + c
the quadratic formula x = b ± b 2 4 a c 2 a
standard form of a quadratic function f ( x ) = a ( x h ) 2 + k

Key concepts

  • A polynomial function of degree two is called a quadratic function.
  • The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
  • The axis of symmetry is the vertical line passing through the vertex. The zeros, or x - intercepts, are the points at which the parabola crosses the x - axis. The y - intercept is the point at which the parabola crosses the y - axis. See [link] , [link] , and [link] .
  • Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See [link] .
  • The vertex can be found from an equation representing a quadratic function. See [link] .
  • The domain of a quadratic function is all real numbers. The range varies with the function. See [link] .
  • A quadratic function’s minimum or maximum value is given by the y - value of the vertex.
  • The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See [link] and [link] .
  • Some quadratic equations must be solved by using the quadratic formula. See [link] .
  • The vertex and the intercepts can be identified and interpreted to solve real-world problems. See [link] .

Section exercises

Verbal

Explain the advantage of writing a quadratic function in standard form.

When written in that form, the vertex can be easily identified.

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How can the vertex of a parabola be used in solving real world problems?

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Explain why the condition of a 0 is imposed in the definition of the quadratic function.

If a = 0 then the function becomes a linear function.

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What is another name for the standard form of a quadratic function?

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What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

If possible, we can use factoring. Otherwise, we can use the quadratic formula.

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Algebraic

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.

f ( x ) = x 2 12 x + 32

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g ( x ) = x 2 + 2 x 3

f ( x ) = ( x + 1 ) 2 2 , Vertex ( 1 , 4 )

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f ( x ) = x 2 + 5 x 2

f ( x ) = ( x + 5 2 ) 2 33 4 , Vertex ( 5 2 , 33 4 )

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h ( x ) = 2 x 2 + 8 x 10

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k ( x ) = 3 x 2 6 x 9

f ( x ) = 3 ( x 1 ) 2 12 , Vertex ( 1 , 12 )

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f ( x ) = 3 x 2 5 x 1

f ( x ) = 3 ( x 5 6 ) 2 37 12 , Vertex ( 5 6 , 37 12 )

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For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

y ( x ) = 2 x 2 + 10 x + 12

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f ( x ) = 2 x 2 10 x + 4

Minimum is 17 2 and occurs at 5 2 . Axis of symmetry is x = 5 2 .

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f ( x ) = x 2 + 4 x + 3

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f ( x ) = 4 x 2 + x 1

Minimum is 17 16 and occurs at 1 8 . Axis of symmetry is x = 1 8 .

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h ( t ) = 4 t 2 + 6 t 1

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f ( x ) = 1 2 x 2 + 3 x + 1

Minimum is 7 2 and occurs at 3. Axis of symmetry is x = 3.

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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