Properties of functions · Grade 9
In Properties of functions (Grade 9) you practise 5 skills: Crossing the axes, positive and negative · Turning points, increasing and decreasing · Solving an equation from the graph · Shifting a graph · Reflecting in the horizontal axis. When you read the graph of a function, look for its properties: where it crosses the axes, where it is above the horizontal axis (there the function is positive) and where it is below (there it is negative), where it rises and where it falls, and its turning points, where it changes direction.
A turning point is a coordinate pair, and its value is the y coordinate: its height. The highest point of the whole graph must be higher than every other point, beyond the picture too. Some graphs have no turning point at all. A graph solves equations: when the function equals a given number, the solutions are the x coordinates of the points where the graph meets the horizontal line at that height. Adding a number to the function moves the graph up or down. A number inside the brackets moves it sideways: a minus moves it right, and a plus moves it left. A minus in front of the function reflects the graph in the horizontal axis, and every value changes sign.
Tip: When the graph carries on beyond the picture, check where its ends are heading before you decide it has a highest point.
What you learn
Crossing the axes, positive and negative
For example:
The picture shows the graph of the function y = f(x) . The graph carries on in the same direction beyond the picture. At which point does the graph cross the horizontal axis?Answer and solution
The answer: (1, 0)
The graph meets the horizontal axis at (1, 0)
Turning points, increasing and decreasing
For example:
The picture shows the graph of the function y = f(x) . The graph carries on in the same direction beyond the picture. What is the maximum point of the function?Answer and solution
The answer: (−3, 3)
The graph rises up to the point (−3, 3) and then falls, so this is the maximum point.
Solving an equation from the graph
For example:
The picture shows the graph of the function y = f(x) . The graph carries on in the same direction beyond the picture. For which values of x is f(x) = 0 ?Answer and solution
The answer: x = 1
The graph crosses the horizontal axis at the point (1, 0) . So x = 1
Shifting a graph
For example: The point (−4, 5) is on the graph of y = f(x) . Which point matches it on the graph of y = f(x) − 3 ?
Answer and solution
The answer: (−4, 2)
The y coordinate goes down by 3, and the x coordinate stays the same: (−4, 2)
Reflecting in the horizontal axis
For example: The point (−4, 5) is on the graph of y = f(x) . Which point matches it on the graph of y = −f(x) ?
Answer and solution
The answer: (−4, −5)
The y coordinate changes sign, and the x coordinate stays the same: (−4, −5)
Not on the site yet
Parts of this topic that are not built yet: Sketching a graph from given properties · The domain of a function · Even and odd functions.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education's grade 9 maths curriculum (2026–27).