Create an algebraic expression based on the following parameters: $$ a=\frac{1}{2},b=\frac{1}{2},c=\frac{1}{2} $$

$$ \frac{x^2}{2}+\frac{x}{2}+\frac{1}{2} $$

$$ \frac{x^2}{2}+2x+\frac{1}{2} $$

$$ \frac{x^2}{2}-\frac{x}{2}-\frac{1}{4} $$

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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Test yourself on plotting functions with parameters!

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

$$ 4x^2+9x-2 $$

Practice more now

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

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

Question 2

Identify the coefficients based on the following equation

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

Question 3

Identify the coefficients based on the following equation

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

Examples with solutions for Plotting Functions with Parameters

Exercise #1

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

$4x^2+9x-2$

Video Solution

Step-by-Step Solution

The quadratic equation is given as $4x^2 + 9x - 2$ . This equation is in the standard form of a quadratic equation, which is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are coefficients.

The term $4x^2$ indicates that the coefficient $a = 4$ .
The term $9x$ indicates that the coefficient $b = 9$ .
The constant term $-2$ indicates that the coefficient $c = -2$ .

From this analysis, we can see that the coefficient $c$ is $-2$ .

Therefore, the value of the coefficient $c$ in the equation is $-2$ .

Answer

-2

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

Identify the coefficients based on the following equation

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

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

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

Exercise #4

Identify the coefficients based on the following equation

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

Video Solution

Step-by-Step Solution

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

Step 1: Identify the given quadratic function.
Step 2: Compare it to the standard form $y = ax^2 + bx + c$ .
Step 3: Determine the values of $a$ , $b$ , and $c$ .

Now, let's work through each step:

Step 1: The problem gives us the quadratic function $y = 3x^2 + 4x + 5$ .

Step 2: The standard form of a quadratic function is $y = ax^2 + bx + c$ .

Step 3: By comparing $y = 3x^2 + 4x + 5$ with $y = ax^2 + bx + c$ , we find:
- The coefficient of $x^2$ is $a = 3$ .
- The coefficient of $x$ is $b = 4$ .
- The constant term is $c = 5$ .

Therefore, the solution to the problem is $a = 3, b = 4, c = 5$ .

This matches choice 2, which states: $a = 3, b = 4, c = 5$ .

Answer

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

Exercise #5

Identify the coefficients based on the following equation

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

Video Solution

Step-by-Step Solution

Let's determine the coefficients for the quadratic function given by $y = -2x^2 + 3x + 10$ .

Step 1: Identify $a$ .
The coefficient of $x^2$ is $-2$ . Thus, $a = -2$ .
Step 2: Identify $b$ .
The coefficient of $x$ is $3$ . Thus, $b = 3$ .
Step 3: Identify $c$ .
The constant term is $10$ . Thus, $c = 10$ .

Comparing these coefficients to the provided choices, the correct answer is:

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

Therefore, the correct choice is Choice 4.

Answer

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

Start practice

Plotting the Quadratic Function Using Parameters a, b and c

Plotting the Quadratic Function

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

Step-by-Step Solution

Answer

More Questions

Plotting Functions with Parameters