Function

Plotting Functions with ParametersZeros of a FuctionPositive and Negative DomainsIncreasing and Decreasing Domain of a ParabolaThe Vertex of the ParabolaSymmetryLinear Function y=mx+bSlopePositivity and NegativityEquation of a Straight LineGraphical RepresentationRepresentation of PhenomenaLinear FunctionThe quadratic functionFunction - Examples, Exercises and Solutions
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The Quadratic Function
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Function

What is a function?

A function is an equation that describes a specific relationship between XX and YY.
Every time we change XX, we get a different YY.

Linear function –

Looks like a straight line, XX is in the first degree.

Quadratic function –

Parabola, XX is in the square.

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Test yourself on the quadratic function!

einstein

For the function in front of you, the slope is?

XY

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Function

What is a function?

A function is an equation that describes a certain relationship between XX and YY.
In every function, XX is the independent variable and YY is the dependent variable. This means that every time we change XX, we get a different YY.
In other words, the YY we get will depend on the XX we substitute into the function.
YY depends on XX and XX does not depend on anything.
An important point: for every XX, there will be only one YY!

For example:
Y=x+5Y = x+5

In this function, if we substitute XX, we get a different YY each time.
Let's check:
Substitute X=1X=1
and we get:
y=1+5y = 1+5
y=6y=6

Substitute X=2X = 2 and we get:
y=7y=7

What is a linear function?

A linear function is a function that looks like a straight line.
It belongs to the family of functions y=mx+by = mx+b where:

aa represents the slope of the function - positive or negative
m>0m>0 - ascending line
m<0m<0 - descending line

bb represents the point where the function intersects the YY axis.

Important Points:

• The function will look like a straight line rising or falling or parallel to the XX axis, but never parallel to the YY axis.
• We will look at the line from left to right.

Let's look at an example of a linear function:

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Test your knowledge
Question 1

For the function in front of you, the slope is?

XY

Question 2

For the function in front of you, the slope is?

XY

Question 3

For the function in front of you, the slope is?

XY

What is a quadratic function?

A quadratic function is a function where X is squared. The quadratic function is also called a second-degree function and is commonly referred to as a parabola.
The basic quadratic function equation is:
Y=ax2+bx+cY = ax^2+bx+c

When -
aa - must be different from 00.

Minimum and maximum parabolas

Minimum Parabola – also called a smiling parabola.
Vertex of the parabola – the point where YY is the smallest.
If aa in the equation is positive (happy) – the parabola is a minimum parabola

Maximum Parabola – A maximum parabola is also called a sad parabola
Vertex of the Parabola – The point where YY is the highest.
If aa in the equation is negative (sad) – the parabola is a maximum parabola

How do you find the vertex of the parabola?

Finding the vertex of the parabola -
The first method: Using the vertex formula of the parabola.

Xvertex=−b2aX_{\text{vertex}} = \frac{-b}{2a}

We substitute aa and bb into the formula from the function's equation and find XX.
After finding the vertex XX, we substitute it into the original function's equation and get the vertex YY.

The second method: Using two symmetrical points
The formula to find XX vertex using two symmetrical points is:

The vertex XX we obtained, we will substitute into the original equation to find the value of the vertex YY.

How do you find the intercepts of the axes with the parabola?

To find the intersection point with the XX axis:
Substitute Y=0Y=0 into the quadratic equation and solve using factoring or the quadratic formula.
To find the intersection point with the YY axis:
Substitute X=0X=0 into the quadratic equation and find the solutions.

Finding the intervals of increase and decrease of a quadratic function based on a graph

Intervals of increase and decrease describe the XX values where the parabola is increasing and where the parabola is decreasing.

Let's see an example:
We will check what happens when the x-values are less than the vertex XX and what happens when the x-values are greater than the vertex XX.
The following quadratic function is given:
It is given that the vertex of the parabola is (−4,−1)(-4, -1) and that the parabola is a minimum parabola.
Solution:
We will draw a sketch according to the data and thus clearly see the increasing and decreasing intervals of the function.

Translation of images:

The value of X at the first point + the value of X at the second point, vertex

Do you know what the answer is?
Question 1

For the function in front of you, the slope is?

XY

Question 2

For the function in front of you, the slope is?

XY

Question 3

For the function in front of you, the slope is?

XY

Examples with solutions for Function

Exercise #1

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To determine the slope of the line shown on the graph, we perform a visual analysis:

  • We examine the orientation of the line from left to right.
  • The red line starts at a higher point on the left and descends to a lower point on the right.
  • This indicates a downward movement, which corresponds to a negative slope.

Therefore, by observing the direction of the line, we conclude that the slope of the function is negative. This positional evaluation confirms that the correct answer is negative slope.

Answer

Negative slope

Exercise #2

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To solve this problem, we need to determine the slope of the line depicted on the graph.

First, understand that the slope of a line on a coordinate plane indicates how steep the line is and the direction it is heading. Specifically:

  • A positive slope means the line rises as it goes from left to right.
  • A negative slope means the line falls as it goes from left to right.

Let's examine the graph given:

  • We see that the line starts at a higher point on the left and descends to a lower point on the right side.
  • As we move from the left side of the graph towards the right, the line goes downwards.

This downward trajectory clearly indicates a negative slope because the line is declining as we move horizontally left to right.

Therefore, the slope of this function is Negative.

The correct answer is, therefore, Negative slope.

Answer

Negative slope

Exercise #3

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To solve this problem, let's analyze the given graph of the function to determine the slope's sign.

The slope of a line on a graph indicates the line's direction. A line with a positive slope rises as it moves from left to right, indicating that for every step taken to the right (along the x-axis), we move upward. Conversely, a line with a negative slope falls as it moves from left to right, meaning each step to the right results in moving downward.

Examining the graph provided, the red line starts higher on the left and goes downward towards the right visually. This indicates that the line is rising as it goes from left to right, which confirms it has a positive slope.

Therefore, the solution to the problem, regarding the slope of the line, is that it is a Positive slope.

Answer

Positive slope

Exercise #4

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

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

  • Step 1: Visual Inspection – Examine the red line on the graph to determine direction.
  • Step 2: Determine Slope Direction – Ascertain if the line rises or falls as it moves from left to right.
  • Step 3: Compare with Possible Answers – Verify which choice aligns with the determined slope direction.

Now, let's work through each step:
Step 1: The graph shows a red line segment, oriented in a manner that moves from left (lower) to right (higher).
Step 2: As the red line moves from the left toward the right side of the graph, it rises, indicating an upward trend and suggesting a positive slope.
Step 3: Given that the line increases from left to right, the slope is positive.

Therefore, the solution to the problem is Positive slope.

Answer

Positive slope

Exercise #5

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To determine the slope of the line segment shown in the graph, follow these steps:

  • Identify the line segment on the graph; it's shown as a red line from one point to another.
  • Examine the direction the line segment travels from the leftmost point to the rightmost point.
  • Visually analyze whether the line segment is rising or falling as it moves from left to right.

Here is the detailed analysis:
- The red line segment starts lower on the left side and ends higher on the right side.
- This suggests that as we move from left to right, the line is rising.
- In terms of slope, a line that rises as it moves from left to right has a positive slope.

Therefore, the slope of the line segment is positive.

Thus, the correct answer is Positive slope.

Answer

Positive slope

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