Plotting the Quadratic Function Using Parameters a, b and c

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:

$a$ – the coefficient of $X^2$ .
$b$ – the coefficient of $X$ .
$c$ – the constant term.

Steps to graph a quadratic function –

Let's examine the parameter $a$ and ask: Is the function upward or downward facing?
Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
Let's find the points of intersection with the $X$ axis by substituting ( $Y=0$ ).
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 $X$ axis that we found. Draw accordingly.

Detailed explanation

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Test yourself on plotting functions with parameters!

$$ y=x^2 $$

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Graphing the quadratic function ausing parameters a, b and c

Let's present the quadratic function and understand the meaning of each parameter in it.
$y=ax^2+bx+c$

$a$ – the coefficient of $X^2$
- -determines if the parabola will be a maximum or minimum parabola (sad or smiling) - determines the steepness – its width.

$a$ must be different from 0.
If $a$ is positive – the parabola is a minimum parabola – smiling
If $a$ is negative – the parabola is a maximum parabola – sad
The larger $a$ is – the narrower the function will be and vice versa.

$b$ - the coefficient of $X$
can be any number
indicates the position of the parabola along with $C$ .
determines the slope of the function at the intersection point with the $Y$ axis
(not relevant to the material)

$c$ – the constant term
is not dependent on $X$
can be any number
indicates the position of the parabola along with $X$
determines the y-intercept
responsible for the vertical shift of the function

Wonderful. Now, let's move on to plotting the quadratic function.

Steps to graph a quadratic function –

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

Let's see an example -

In the following function:
$Y=-4X^2+4x+3$

$a=-4$ , negative. Therefore, the function is downward facing.
Let's find the vertex of the parabola:
$X=\frac{-4}{-4*2}=\frac{-4}{-8}=\frac{1}{2}$
$y=-4*\frac{1^2}{2}+4*\frac{1}{2}+3$
$y=4$
Vertex of the parabola $(\frac{1}{2}, 4)$
Let's find the intersection points with the $X$ axis
Substitute
$Y=0$
and we get:
$-4X^2+4x+3=0$
$X=-\frac{1}{2},1\frac{1}{2}$
Draw a coordinate system, mark relevant points, and draw logically:

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Test your knowledge

Question 1

$$ y=x^2+10x $$

Question 2

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

Question 3

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

Examples with solutions for Plotting Functions with Parameters

Exercise #1

$y=x^2+10x$

Video Solution

Step-by-Step Solution

Here we have a quadratic equation.

A quadratic equation is always constructed like this:

$y = ax²+bx+c$

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.

$c = 0$

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

$a = 1$

$b= 10$

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

Answer

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

Exercise #2

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

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

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

Exercise #3

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

$3x^2+8x-5$

Video Solution

Step-by-Step Solution

The quadratic equation of the given problem is already arranged (that is, all the terms are found on one side and the 0 on the other side), thus we approach the given problem as follows;

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

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

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

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

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

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

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

$x$ is

$8$ Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number $8$ ,

Thus the correct answer is option d.

Answer

Exercise #4

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

$3x^2+5x$

Video Solution

Step-by-Step Solution

The quadratic equation of the given problem has already been arranged (that is, all the terms are on one side and 0 is on the other side) thus we can approach the question as follows:

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

Let's remember the definition of a coefficient when solving a quadratic equation as well as the formula for the roots:

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

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

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

That is the coefficient
$c$ is the free term - and as such the coefficient of the term is raised to the power of zero - $x^0$ (Any number other than zero raised to the power of zero equals 1:

$x^0=1$ )

Next we examine the equation of the given problem:

$3x^2+5x=0$ 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:

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

Hence the correct answer is option c.

Answer

Exercise #5

$y=x^2$

Video Solution

Answer

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

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Plotting the Quadratic Function Using Parameters a, b and c

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:

Steps to graph a quadratic function –

Test yourself on plotting functions with parameters!

Graphing the quadratic function ausing parameters a, b and c

Steps to graph a quadratic function –

Examples with solutions for Plotting Functions with Parameters

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Answer