Parameter functions and graph plotting - Examples, Exercises and Solutions

Plotting the graph of the quadratic function and examining the roles of the parameters \(a, b, c\) in the function of the form \(y = ax^2 + bx + c\)

The quadratic function has three relevant characteristics:

aa – the coefficient of X2X^2.
bb – the coefficient of XX.
cc – the constant term.

Steps to graph a quadratic function –

  1. Let's examine the parameter aa and ask: Is the function upward or downward facing?
  2. Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
  3. Let's find the points of intersection with the XX axis by substituting (Y=0Y=0).
  4. Let's draw a coordinate system and first mark the vertex of the parabola.
    Then, let's examine if the function is smiling or crying and mark the points of intersection with the XX axis that we found. Draw accordingly.

Suggested Topics to Practice in Advance

  1. The quadratic function
  2. Parabola

Practice Parameter functions and graph plotting

Exercise #1

What is the value of the coefficient b b in the equation below?

3x2+8x5 3x^2+8x-5

Video Solution

Step-by-Step Solution

The quadratic equation of the problem is already arranged (that is, all the terms on one side and 0 on the other side), so we approach answering the question posed:

In the problem, the question was asked: what is the value of the coefficientb b in the equation?

Let's remember the definitions of the coefficients when solving a quadratic equation and the formula for the roots:

The rule says that the roots of an equation of the form

ax2+bx+c=0 ax^2+bx+c=0 are :

x1,2=b±b24ac2a x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

That is the coefficientb b is the coefficient of the term in the first power -x x We examine the equation of the problem:

3x2+8x5=0 3x^2+8x-5 =0 That is, the number that multiplies

x x is

8 8 And then we recognize b, which is the coefficient of the term in the first power, is the number8 8 ,

The correct answer is option d.

Answer

8

Exercise #2

y=x2+10x y=x^2+10x

Video Solution

Step-by-Step Solution

Here we have a quadratic equation.

A quadratic equation is always constructed like this:

 

y=ax2+bx+c y = ax²+bx+c

 

Where a, b, and c are generally already known to us, and the X and Y points need to be discovered.

Firstly, it seems that in this formula we do not have the C,

Therefore, we understand it is equal to 0.

c=0 c = 0

 

a is the coefficient of X², here it does not have a coefficient, therefore

a=1 a = 1

 

b=10 b= 10

is the number that comes before the X that is not squared.

 

Answer

a=1,b=10,c=0 a=1,b=10,c=0

Exercise #3

y=2x25x+6 y=2x^2-5x+6

Video Solution

Step-by-Step Solution

In fact, a quadratic equation is composed as follows:

y = ax²-bx-c

 

That is,

a is the coefficient of x², in this case 2.
b is the coefficient of x, in this case 5.
And c is the number without a variable at the end, in this case 6.

Answer

a=2,b=5,c=6 a=2,b=-5,c=6

Exercise #4

What is the value of the coefficient c c in the equation below?

3x2+5x 3x^2+5x

Video Solution

Step-by-Step Solution

The quadratic equation of the problem is already ordered (that is, all the terms on one side and 0 on the other side), so we approach answering the question posed:

In the problem, the question was asked: what is the value of the coefficientc c in the equation?

Let's remember the definitions of the coefficients when solving a quadratic equation and the formula for the roots:

The rule says that the roots of an equation of the form

ax2+bx+c=0 ax^2+bx+c=0 are:

x1,2=b±b24ac2a x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

That is the coefficient
c c is the free term - that is, the coefficient of the term raised to the power of zero -x0 x^0 (And this is because any number other than zero raised to the power of zero equals 1:

x0=1 x^0=1 )

We examine the equation of the problem:

3x2+5x=0 3x^2+5x=0 Note that there is no free term in the equation, that is, the numerical value of the free term is 0, in fact the equation can be written as follows:

3x2+5x+0=0 3x^2+5x+0=0 and therefore the value of the coefficientc c is 0.

The correct answer is option c.

Answer

0

Exercise #5

What is the value ofl coeficiente a a in the equation?

x2+7x9 -x^2+7x-9

Video Solution

Answer

-1

Exercise #1

What is the value of the coefficient c c in the equation below?

4x2+9x2 4x^2+9x-2

Video Solution

Answer

-2

Exercise #2

y=x2 y=x^2

Video Solution

Answer

a=1,b=0,c=0 a=1,b=0,c=0

Exercise #3

y=x26x+4 y=x^2-6x+4

Video Solution

Answer

a=1,b=6,c=4 a=1,b=-6,c=4

Exercise #4

y=2x23x6 y=2x^2-3x-6

Video Solution

Answer

a=2,b=3,c=6 a=2,b=-3,c=-6

Exercise #5

y=2x2+3x+10 y=-2x^2+3x+10

Video Solution

Answer

a=2,b=3,c=10 a=-2,b=3,c=10

Exercise #1

y=3x2+4x+5 y=3x^2+4x+5

Video Solution

Answer

a=3,b=4,c=5 a=3,b=4,c=5

Exercise #2

What is the value of the coefficient b b in the equation below?

x2=2x+7 x^2=2x+7

Video Solution

Answer

∓2

Exercise #3

y=x2+x+5 y=x^2+x+5

Video Solution

Answer

a=1,b=1,c=5 a=1,b=1,c=5

Exercise #4

y=x2+x+5 y=-x^2+x+5

Video Solution

Answer

a=1,b=1,c=5 a=-1,b=1,c=5

Exercise #5

y=4+3x2x y=4+3x^2-x

Video Solution

Answer

a=3,b=1,c=4 a=3,b=-1,c=4

Topics learned in later sections

  1. Finding the Zeros of a Parabola
  2. Vertex of a parabola
  3. Symmetry in a parabola