Plotting the Quadratic Function Using Parameters a, b and c

Plotting the Quadratic Function

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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$$ y=x^2 $$

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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

$y=x^2$

Video Solution

Step-by-Step Solution

To solve this problem, let's follow these steps:

Step 1: Recognize the standard form of a quadratic equation.
Step 2: Match the given function to the standard form.
Step 3: Identify each coefficient $a$ , $b$ , and $c$ .

Now, let's work through these steps:

Step 1: The standard form of a quadratic function is $y = ax^2 + bx + c$ . Our goal is to identify $a$ , $b$ , and $c$ .

Step 2: We are given the function $y = x^2$ . This can be aligned with the standard form as $y = 1 \cdot x^2 + 0 \cdot x + 0$ .

Step 3: By comparing the given function $y = x^2$ with the standard form, we can deduce:
- The coefficient of $x^2$ is 1, so $a = 1$ .
- The linear term coefficient is missing, which implies $b = 0$ .
- There is no constant term, so $c = 0$ .

Therefore, the coefficients are $a = 1, b = 0, c = 0$ , corresponding to choice 1.

Answer

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

Exercise #2

$y=x^2+10x$

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 #3

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

Video Solution

Step-by-Step Solution

To solve this problem, we'll clearly delineate the given expression and compare it to the standard quadratic form:

Step 1: Recognize the standard form of a quadratic equation as $y = ax^2 + bx + c$ .
Step 2: Compare the given equation $y = x^2 - 6x + 4$ to the standard form.
Step 3: Identify coefficients:
- The coefficient of $x^2$ is $a = 1$ .
- The coefficient of $x$ is $b = -6$ .
- The constant term is $c = 4$ .

Therefore, the coefficients for the quadratic function $y = x^2 - 6x + 4$ are $a = 1$ , $b = -6$ , and $c = 4$ .

Among the provided choices, choice 3: $a=1,b=-6,c=4$ is the correct one.

Answer

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

Exercise #4

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

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 #5

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

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Identify the given quadratic function.
Match it with the standard form of a quadratic equation $y = ax^2 + bx + c$ .
Extract the values of $a$ , $b$ , and $c$ directly from the comparison.

Now, let's work through each step:
Step 1: The given quadratic function is $y = 2x^2 - 3x - 6$ .
Step 2: The standard form of a quadratic equation is $y = ax^2 + bx + c$ .
Step 3: By matching the given quadratic function with the standard form:

- The coefficient of $x^2$ is $2$ , so $a = 2$ .
- The coefficient of $x$ is $-3$ , so $b = -3$ .
- The constant term is $-6$ , so $c = -6$ .

Therefore, the solution to the problem is $a = 2$ , $b = -3$ , $c = -6$ .

Answer

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

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Plotting the Quadratic Function Using Parameters a, b and c