The standard representation of the quadratic function looks like this:
The standard representation of the quadratic function looks like this:
When:
–
-
–
Let's see an example:
We are given the function:
What can we conclude?
therefore the parabola is smiling
therefore the function intersects the -axis at the point
Click here to learn more about the standard form of a quadratic function
The vertex form of a quadratic function allows us to identify its vertex directly from the function!
The vertex form of a quadratic function is:
When:
- represents the value of the vertex.
- represents the value of the vertex.
For example:
In the function
The vertex of the parabola is:
Note-
The vertex formula is structured such that there is always a – before the , meaning vertex, but this does not necessarily mean that the vertex is negative.
If the parabola has a negative vertex, a will appear before the in the formula because times equals .
For example:
For example:
In the function
The vertex of the parabola is:
There is a before the in the formula, so it is .
Click here to learn more about the vertex form of a quadratic function
The product form shows multiplication between expressions. With the product form, we can easily determine the points of intersection of the function with the -axis.
The product form of the quadratic function looks like this:
where
and are the points of intersection of the parabola with the -axis.
As follows:
Let's see an example of the product form to understand better:
We can determine that:
The points of intersection with the -axis are:
Note - Since in the original template there is a minus before \(k\) and , we can infer that if there is a plus before one of them, it is negative, hence and not .
Click here to learn more about representing as a product of a quadratic function
Sometimes we encounter quadratic equations that are missing terms such as or and quadratic equations with denominators.
Quadratic equations with missing terms are quadratic equations where or are equal to .
When we have an incomplete equation where b=0:
We equate the constant term to the term with
and solve for . Note that a square root has two answers (negative and positive).
Let's see an example:
We equate the constant term to
and get:
When we have an incomplete equation where :
we can immediately determine that the parabola intersects the axis when
we will factor out the common factor and find the roots of the equation.
For example:
We factor out a common factor and get:
The factors that zero the equation are –
Sometimes we encounter quadratic equations with a fraction (numerator and denominator). To read the function more accurately, we need to get rid of the fraction.
To solve quadratic equations with denominators-
find the common denominator, multiply each term, and obtain an equation without a fraction. Then solve it completely normally and find the solutions.
Click here to learn more about different representations of a quadratic function
Create an algebraic expression based on the following parameters:
\( a=-1,b=-1,c=-1 \)
Create an algebraic expression based on the following parameters:
\( a=-1,b=-2,c=-5 \)
Create an algebraic expression based on the following parameters:
\( a=4,b=2,c=5 \)
Create an algebraic expression based on the following parameters:
\( a=-1,b=1,c=0 \)
Create an algebraic expression based on the following parameters:
\( a=1,b=1,c=0 \)
Create an algebraic expression based on the following parameters:
The goal is to express the quadratic equation using the given parameters , , and .
First, substitute the values of , , and into the standard form:
Combine these terms to form the full expression:
Therefore, the algebraic expression for the parameters , , and is: .
Comparing with the given choices, the correct choice is option 4:
Create an algebraic expression based on the following parameters:
To create the algebraic expression for the quadratic function given the parameters, we follow these steps:
Substituting these values, we get:
Simplify this expression:
This simplifies to .
Therefore, the algebraic expression is .
Create an algebraic expression based on the following parameters:
To derive the algebraic expression based on the parameters given, we follow these steps:
Now, let's implement these steps to form the quadratic expression:
Step 1: The given parameters are , , and .
Step 2: Our basis is the quadratic form .
Step 3: Substituting the given values, we find:
This substitution provides us with the quadratic expression , fulfilling the problem's requirements.
Therefore, the correct algebraic expression is .
Create an algebraic expression based on the following parameters:
To determine the algebraic expression, we start with the standard quadratic function:
Given the values:
We substitute these into the formula:
Simplifying the expression gives:
Thus, the algebraic expression, when these parameters are substituted, is:
The solution to the problem is .
Create an algebraic expression based on the following parameters:
To determine the algebraic expression, we will substitute the given parameters into the standard form of the quadratic function:
Substituting these values, the expression becomes:
.
This simplifies to:
.
Therefore, the algebraic expression, based on the given parameters, is .
Create an algebraic expression based on the following parameters:
\( a=3,b=0,c=-3 \)
Create an algebraic expression based on the following parameters:
\( a=1,b=-1,c=1 \)
Create an algebraic expression based on the following parameters:
\( a=1,b=16,c=64 \)
Create an algebraic expression based on the following parameters:
\( a=4,b=-16,c=0 \)
Create an algebraic expression based on the following parameters:
\( a=\frac{1}{2},b=\frac{1}{2},c=\frac{1}{2} \)
Create an algebraic expression based on the following parameters:
To solve the problem of creating an algebraic expression with the given parameters, we will proceed as follows:
Through substitution, the expression becomes:
We can further simplify this expression:
Thus, the algebraic expression with the given parameters is .
The correct answer corresponds to choice number 1: .
Therefore, the solution to the problem is
Create an algebraic expression based on the following parameters:
To solve this problem, let's form the algebraic expression using the standard quadratic formula:
Given are the values:
,
,
.
Substituting these values into the formula, we have:
This simplifies to:
Thus, the algebraic expression is .
The correct choice from the given options is:
Choice 3:
Create an algebraic expression based on the following parameters:
To solve this problem, let's proceed with the construction of the quadratic expression:
Thus, the algebraic expression we derive from these parameters is the quadratic expression:
This matches the correct choice provided in the given multiple-choice options.
Create an algebraic expression based on the following parameters:
To solve this problem, we'll follow these steps:
Now, let's work through each step:
Step 1: We use the standard form of a quadratic expression, which is .
Step 2: Substitute the values , , and into this template:
Step 3: Simplify the expression:
The expression simplifies to .
Thus, the algebraic expression based on the given parameters is .
Checking against the answer choices, the correct choice is:
Create an algebraic expression based on the following parameters:
To solve this problem, we'll follow these steps:
Let's work through each step:
Step 1: The given coefficients are , , and . Substitute these values into the standard quadratic form :
Step 2: The expression is already simplified. The coefficients are correctly substituted, and no further simplification is needed:
Step 3: Compare this expression to the provided multiple-choice options. The correct match is:
Choice 1:
Therefore, the algebraic expression is .
Create an algebraic expression based on the following parameters:
\( a=5,b=3,c=-4 \)
Create an algebraic expression based on the following parameters:
\( a=1,b=2,c=0 \)
Create an algebraic expression based on the following parameters:
\( a=1,b=-1,c=3 \)
Create an algebraic expression based on the following parameters:
\( a=3,b=0,c=0 \)
Create an algebraic expression based on the following parameters:
\( a=2,b=\frac{1}{2},c=4 \)
Create an algebraic expression based on the following parameters:
To solve this problem, we'll follow these steps:
Now, let's work through each step:
Step 1: The general form of a quadratic expression is .
Step 2: We are given , , and . Substituting these into the expression, we get:
Therefore, the solution to the problem is .
Create an algebraic expression based on the following parameters:
To solve this problem, we'll follow these steps:
Now, let's work through each step:
Step 1: We are given , , and .
Step 2: We'll use the formula to form our expression.
Step 3: By substituting the given values, we get:
Therefore, we combine the terms to form the expression: .
The correct answer choice based on our derived expression is: .
Create an algebraic expression based on the following parameters:
To solve this problem, we'll follow these steps:
Now, let's perform these steps:
Step 1: The problem provides us with the coefficients , , and for a quadratic expression .
Step 2: Substitute these values into the quadratic expression:
: Multiply by , resulting in .
: Multiply by , resulting in .
: The constant term is .
Thus, the algebraic expression is:
.
Comparing this result to the given choices, we find that this expression matches choice 3.
Therefore, the solution to the problem is .
Create an algebraic expression based on the following parameters:
To solve this problem, we'll follow these steps:
Let's execute these steps:
Step 1: Substitute the values into the formula:
Step 2: Simplify the expression:
Eliminate the terms with zero coefficients to get:
Thus, the algebraic expression for the quadratic function with , , and is .
Therefore, the correct choice from the options provided is choice 1:
Create an algebraic expression based on the following parameters:
To solve this problem, we'll follow the steps outlined:
Now, let's proceed with these steps:
Given the standard form of a quadratic expression :
Substituting the values, we obtain:
Therefore, the correct algebraic expression for the quadratic function is .