A function is an equation that describes a specific relationship between and .
Every time we change , we get a different .
A function is an equation that describes a specific relationship between and .
Every time we change , we get a different .
Looks like a straight line, is in the first degree.
Parabola, is in the square.
For the function in front of you, the slope is?
A function is an equation that describes a certain relationship between and .
In every function, is the independent variable and is the dependent variable. This means that every time we change , we get a different .
In other words, the we get will depend on the we substitute into the function.
depends on and does not depend on anything.
An important point: for every , there will be only one !
For example:
In this function, if we substitute , we get a different each time.
Let's check:
Substitute
and we get:
Substitute and we get:
A linear function is a function that looks like a straight line.
It belongs to the family of functions where:
represents the slope of the function - positive or negative
- ascending line
- descending line
represents the point where the function intersects the axis.
Important Points:
• The function will look like a straight line rising or falling or parallel to the axis, but never parallel to the axis.
• We will look at the line from left to right.
Let's look at an example of a linear function:
For the function in front of you, the slope is?
For the function in front of you, the slope is?
For the function in front of you, the slope is?
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:
When -
- must be different from .
Minimum Parabola – also called a smiling parabola.
Vertex of the parabola – the point where is the smallest.
If 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 is the highest.
If in the equation is negative (sad) – the parabola is a maximum parabola
Finding the vertex of the parabola -
The first method: Using the vertex formula of the parabola.
We substitute and into the formula from the function's equation and find .
After finding the vertex , we substitute it into the original function's equation and get the vertex .
The second method: Using two symmetrical points
The formula to find vertex using two symmetrical points is:
The vertex we obtained, we will substitute into the original equation to find the value of the vertex .
To find the intersection point with the axis:
Substitute into the quadratic equation and solve using factoring or the quadratic formula.
To find the intersection point with the axis:
Substitute into the quadratic equation and find the solutions.
Intervals of increase and decrease describe the 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 and what happens when the x-values are greater than the vertex .
The following quadratic function is given:
It is given that the vertex of the parabola is 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
For the function in front of you, the slope is?
For the function in front of you, the slope is?
For the function in front of you, the slope is?
For the function in front of you, the slope is?
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.
Negative slope
What is the solution to the following inequality?
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.
We start by organizing the sections:
Divide by 13 to isolate the X
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 than, 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 to, 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!
Look at the linear function represented in the diagram.
When is the function positive?
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:
meaning any number greater than 2:
x > 2
x>2
Look at the function shown in the figure.
When is the function positive?
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
-4 > x
For the function in front of you, the slope is?
To determine the slope of the line segment shown in the graph, follow these steps:
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.
Positive slope