Plotting Functions with Parameters - Examples, Exercises and Solutions

Understanding Plotting Functions with Parameters

Complete explanation with examples

Plotting the Quadratic Function

Plotting the graph of the quadratic function and examining the roles of the parameters a,b,ca, b, c in the function of the form y=ax2+bx+cy = 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.
Detailed explanation

Practice Plotting Functions with Parameters

Test your knowledge with 18 quizzes

\( y=-x^2+x+5 \)

Examples with solutions for Plotting Functions with Parameters

Step-by-step solutions included
Exercise #1

Determine the value of the coefficient a a in the following equation:

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

Step-by-Step Solution

The quadratic equation in the problem is already arranged (meaning all terms are on one side and 0 on the other side), so let's proceed to answer the question asked:

The question asked in the problem - What is the value of the coefficienta a in the equation?

Let's recall the definitions of coefficients in solving quadratic equations and the roots formula:

The rule states 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 a a is the coefficient of the quadratic term (meaning the term with the second power)- x2 x^2 Let's examine the equation in the problem:

x2+7x9=0 -x^2+7x-9 =0

Remember that the minus sign before the quadratic term means multiplication by: 1 -1 , therefore- we can write the equation as:

1x2+7x9=0 -1\cdot x^2+7x-9 =0

The number that multiplies the x2 x^2 , is 1 -1 hence we identify that the coefficient of the quadratic term is the number 1 -1 ,

Therefore the correct answer is A.

Answer:

-1

Video Solution
Exercise #2

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

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

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 coefficientb 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

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 then examine the equation of the given problem:

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

x x is

8 8 Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number8 8 ,

Thus the correct answer is option d.

Answer:

8

Video Solution
Exercise #3

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

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

Step-by-Step Solution

The quadratic equation is given as 4x2+9x2 4x^2 + 9x - 2 . This equation is in the standard form of a quadratic equation, which is ax2+bx+c ax^2 + bx + c , where a a , b b , and c c are coefficients.

  • The term 4x2 4x^2 indicates that the coefficient a=4 a = 4 .
  • The term 9x 9x indicates that the coefficient b=9 b = 9 .
  • The constant term 2-2 indicates that the coefficient c=2 c = -2 .

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

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

Answer:

-2

Video Solution
Exercise #4

y=x2 y=x^2

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 a , b b , and c c .

Now, let's work through these steps:

Step 1: The standard form of a quadratic function is y=ax2+bx+c y = ax^2 + bx + c . Our goal is to identify a a , b b , and c c .

Step 2: We are given the function y=x2 y = x^2 . This can be aligned with the standard form as y=1x2+0x+0 y = 1 \cdot x^2 + 0 \cdot x + 0 .

Step 3: By comparing the given function y=x2 y = x^2 with the standard form, we can deduce:
- The coefficient of x2 x^2 is 1, so a=1 a = 1 .
- The linear term coefficient is missing, which implies b=0 b = 0 .
- There is no constant term, so c=0 c = 0 .

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

Answer:

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

Video Solution
Exercise #5

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

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

Video Solution

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