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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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?

Examples with solutions for Function

Exercise #1

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

XY

Video Solution

Step-by-Step Solution

For this problem, we need to determine the nature of the slope for a given straight line on a graph.

Based on the graph provided, the red line starts at a higher point on the left (Y-axis) and moves downward toward a lower point on the right (X-axis). This indicates that as one moves from left to right across the graph, the function decreases in value. Consequently, this is typical of a line that has a negative slope.

The slope of a line is typically defined as the "rise over the run," or the ratio of the change in the vertical direction to the change in the horizontal direction. Here, as we proceed from left to right, the line goes "downwards" (negative rise), establishing a negative slope.

Thus, we can conclude that the slope of the line is negative.

Therefore, the solution to the problem is Negative slope.

Answer

Negative slope

Exercise #2

What is the solution to the following inequality?

10x43x8 10x-4≤-3x-8

Video Solution

Step-by-Step Solution

In the exercise, we have an inequality equation.

We treat the inequality as an equation with the sign -=,

And we only refer to it if we need to multiply or divide by 0.

 10x43x8 10x-4 ≤ -3x-8

We start by organizing the sections:

10x+3x48 10x+3x-4 ≤ -8

13x48 13x-4 ≤ -8

13x4 13x ≤ -4

Divide by 13 to isolate the X

x413 x≤-\frac{4}{13}

Let's look again at the options we were asked about:

Answer A is with different data and therefore was rejected.

Answer C shows a case where X is greater than413 -\frac{4}{13} , although we know it is small, so it is rejected.

Answer D shows a case (according to the white circle) where X is not equal to413 -\frac{4}{13} , and only smaller than it. We know it must be large and equal, so this answer is rejected.

 

Therefore, answer B is the correct one!

Answer

Exercise #3

Look at the linear function represented in the diagram.

When is the function positive?

–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333000

Video Solution

Step-by-Step Solution

The function is positive when it is above the X-axis.

Let's note that the intersection point of the graph with the X-axis is:

(2,0) (2,0) meaning any number greater than 2:

x > 2

Answer

x>2

Exercise #4

Look at the function shown in the figure.

When is the function positive?

xy-4-7

Video Solution

Step-by-Step Solution

The function we see is a decreasing function,

Because as X increases, the value of Y decreases, creating the slope of the function.

We know that this function intersects the X-axis at the point x=-4

Therefore, we can understand that up to -4, the values of Y are greater than 0, and after -4, the values of Y are less than zero.

Therefore, the function will be positive only when

X < -4

 

Answer

-4 > x

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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