What do all linear functions of the form f(x) = ax + a, a > 0, have in common?
Step-by-step solution
Idea: Factorise: ax + a = a(x + 1). Whatever a is, putting x = −1 makes the bracket 0.
- f(x) = ax + a = a(x + 1). At x = −1: f(−1) = a × 0 = 0 for every a. So all the graphs pass through the point (−1, 0): they all cut the x-axis there.1½ marks
- In y = ax + b form, the slope is a and the y-intercept is also a: each line cuts the y-axis at (0, a), which is above the origin because a > 0.1 mark
- Since the slope a is positive, every such line rises from left to right (linear growth). For example y = x + 1, y = 2x + 2 and y = 3x + 3 all meet at (−1, 0) (see the graph).½ mark
Check: a = 5: f(−1) = −5 + 5 = 0 ✓. a = 14: f(−1) = −¼ + ¼ = 0 ✓.
Answer to write in the exam
f(x) = ax + a = a(x + 1)
f(−1) = a(0) = 0 for every a
Slope = a > 0, y-intercept = a > 0
∴ All such lines pass through (−1, 0), rise from left to right, and have slope equal to their y-intercept
Common mistakes that cost marks
- Saying they all pass through the origin. f(0) = a, which is not 0.
- Saying they are parallel. Different values of a give different slopes.
- Testing only one or two values of a and not explaining why it works for all, which the factorised form a(x + 1) shows.
How this can come in the exam
All lines of the form y = kx − 2k pass through the point
- (0, 0)
- (2, 0)
- (−2, 0)
- (0, 2)
Show answer
(B) (2, 0)
y = k(x − 2), which is 0 when x = 2.
Assertion (A): The graphs of y = 4x + 4 and y = 7x + 7 meet on the x-axis.
Reason (R): Both equal a(x + 1), which is 0 at x = −1.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
Both are 0 at x = −1, so both pass through (−1, 0) on the x-axis. R explains A.
Try one yourself
What do all lines y = ax + 3a (a > 0) have in common?
Show answer
y = a(x + 3), so all pass through (−3, 0); each rises to the right, and its y-intercept 3a is 3 times its slope.
More questions like this
- Raju went to a shop where there were sealed boxes of different colours on sale. The shop owner told him that the red boxes have 4 pens each and the blue boxes have 5 pencils each. Now, if Raju bought x red boxes and y blue boxes, how can he quickly figure out the total quantity of pens and pencils? Also, if he got 3 extra pens free, how many pens and pencils did he get altogether?
- A rectangular garden of length l metres and width w metres has to be fenced and decorated. A wire fence is to be laid along the length costing ₹100 per metre and a wooden fence is to be built along the width costing ₹80 per metre. Special seeds have to be sown throughout the garden which will cost ₹50 per square metre. What will be the total cost incurred?
- Thus, 200l + 160w + 50lw is the algebraic expression for the total cost.
1. Can you identify the terms, variables and coefficients of this algebraic expression?
2. How is it different from the algebraic expression 4x + 5y + 3 for Raju’s pens and pencils? - A wire of length 20 cm is bent in different ways to form rectangles. For example, we can have a rectangle with length 7 cm and width 3 cm. We can also have one of length 5.5 cm and width 4.5 cm. (Think of a few more ways of forming such rectangles.) Can you write an expression for the area of such rectangles?
- The expression for the area of these rectangles is x(10 − x) or 10x − x2.
1. Can you identify the terms, variables and coefficients of this algebraic expression?
2. Can you point out any similarity or difference between the algebraic expressions obtained for Raju’s pens and pencils (4x + 5y + 3) and for these rectangles?