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mathematics UK Year 8 Mathematics (KS3)

Solved problems for the chapter:

Introduction to Linear Functions

The chapter does not contain theoretical materials.

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Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:introduction_to_linear_functions/4092/

Example 1.

For the given x, calculate the value of the function f:

at .

Example 2.

Show that the lines and intersect on the x-axis.

For the given x, calculate the value of the function f:

at .

Given the function f (x) = 0.5x + 2. Calculate the value of the function for the given value of x below.

The function is given. Calculate for which value of the variable , for which applies.

Draw the graph of the function, given below:

The graph of function intersects the y-axis at point .

Write the expression for the function and draw its graph.

10.

Draw a graph of the function. Do points and lie on the graph of the function f?

11.

Determine the slope of the linear function so that its graph will pass through the point . Write down the function.

12.

Write down the function g, which is parallel to the graph of the function and passes through the point .

13.

In the coordinate system, draw a line that is parallel to the bisectors of the odd quadrants and intersects the y- axis at . Write down the equation of the line thus obtained .

14.

The figure shows the graph of the functions f and g. Write down their equations.

15.

Write down the zero of the function

16.

Write down the zero of the function

17.

Write down the zero of the function:

18.

Write down the zero of the function .

19.

Draw a graph of the function .

20.

Draw the graph of the function .

21.

Draw a line with the equation:

and calculate the angle of inclination to the nearest minute.

22.

For what value of x is the function positive?

23.

For what value of x is the function negative?

24.

Calculate the area of a triangle bounded by the coordinate axes of the function:

25.

In the same coordinate system draw a graph of the functions and

Calculate the point at which the graphs f and g intersect.

26.

The function is given:

Draw the graph of the function.
Obtain the intersections of the function f with the bisector of the odd quadrants.
Write down the range of the function.

27.

A car with a constant speed of drove past us on a flat road within time t.

Write down the change in the distance (s) traveled of the car as a function of time and present this dependence graphically.
Calculate the distance of the car from us after seconds.
Write down the change in speed as a function of time and present the dependence graphically.

28.

Draw a graph of the function

29.

Draw the graph of the function

30.

The function is given

Write the function f with linear functions at appropriately selected intervals.
Calculate the zeros of the function f.
Draw a graph of the function f.
Calculate the intersection of the function f with the line with the equation

31.

A function is given Obtain the intersections of the graph of the function f with the coordinate axes and draw its graph.

32.

Calculate the zero and initial value of the function and draw its graph. Write the interval on which the function is negative.

33.

The function f is given by the rule:

Draw the graph and calculate the area of the figure between the graph and the abscissa axis on the interval .

34.

Write the equation of the line p passing through the points and . Determine the ordinate of the point so that it lies on the line p.

35.

Calculate the distance of the line from the coordinate origin.

36.

A line with the equation is given. Calculate the intersections of the line with the coordinate axes and draw the line in the coordinate system.

37.

The functions are given below:

Draw the graph of both functions on the same coordinate system.
Calculate the acute angle formed at the intersection of the functions f(x) and g(x). Write down the calculated angle to the nearest hundredth of a second.
Calculate at what value of x the function f(x) is 10 greater value than the function g(x).

38.

In the coordinate system, draw the line given by the equation

Check mathematically whether the point lies on this line.

39.

The lines are given

Draw the lines in the coordinate system and describe their relative position.

40.

A straight line with the equation is given:

Write down the coordinates of the intersections of the given line with the coordinate axes and draw it in the coordinate system.
Write down the equation of the line that is parallel to the given line and passes through the point A(0,3).
Calculate and record the intersection of the given line with the line:

41.

Draw the graph of the equation below:

Write down the coordinates of the intersections of the line with the coordinate axes.

42.

Write the equation of the line passing through the point A (1,0) and parallel to the line in the graph below.

43.

We start filling the pool with water. When filling began, the water in the pool had already reached a certain height. The water level increases linearly with time. The figure shows the graph of the function f, which shows the change in the height h of the water in the pool depending on the time t. Answer the questions below. Water height is measured in meters and time in minutes.

What was the height of the water in the pool when it started filling?
What was the height of the water in the pool one hour after it started filling?
How much did the water level in the pool increase every 15 minutes?
Write down the function f.