Graphs of functions

1. For y = 2x - 3, what is y when x = 4?

Answer A: 5

y = 2(4) - 3 = 8 - 3 = 5, so the coordinate (4, 5) lies exactly on the line.

2. A table is being constructed for y = x^2 - x - 2. Which y-value belongs to x = -2?

Answer B: 4

At x = -2, y = (-2)^2 - (-2) - 2 = 4 + 2 - 2 = 4. The table therefore contains (-2, 4).

3. A straight line passes through (-1, -5) and (3, 7). What is its gradient?

Answer C: 3

The gradient is [7 - (-5)]/[3 - (-1)] = 12/4 = 3. Both differences use the same coordinate order.

4. What is the y-intercept of the line y = -2x + 6?

Answer D: (0, 6)

Substituting x = 0 gives y = 6, so the line crosses the y-axis at the coordinate (0, 6).

5. The graph of y = x^2 - 5x + 6 crosses the x-axis at exactly (2, 0) and (3, 0). What are its roots?

Answer A: x = 2 or 3

The stated x-axis intersections have x-coordinates 2 and 3, so the roots are x = 2 and x = 3.

6. For y = (x + 1)^2 - 4, which coordinate is the minimum turning point?

Answer B: (-1, -4)

The squared term is smallest when x + 1 = 0, so x = -1. Then y = -4, giving the minimum (-1, -4).

7. A quadratic has roots at x = 1 and x = 5. What is the equation of its vertical line of symmetry?

Answer C: x = 3

The midpoint of 1 and 5 is (1 + 5)/2 = 3, so the vertical line of symmetry is x = 3.

8. For y = x^3 - 4x, which list gives all x-axis intersection coordinates?

Answer D: (-2, 0), (0, 0), (2, 0)

x^3 - 4x = x(x - 2)(x + 2), so y = 0 at x = -2, 0 and 2. The intersections are (-2, 0), (0, 0) and (2, 0).

9. For y = -2x^3, what happens to y as x becomes a large positive number?

Answer A: y becomes a large negative number

For large positive x, x^3 is large and positive. Multiplication by -2 makes y large and negative.

10. Which vertical asymptote belongs to y = 8/x?

Answer B: x = 0

The rule is undefined at x = 0, and the branches approach that vertical line. Therefore the vertical asymptote is x = 0.

11. Which coordinate lies on y = 8/x when x = -4?

Answer C: (-4, -2)

y = 8/(-4) = -2, so the ordered pair is (-4, -2). Its coordinate product is 8, confirming the point.

12. The points (-3, 1), (-1, 9), (1, 9) and (3, 1) lie on y = 9/x^2. Which symmetry do these coordinates show?

Answer D: Symmetry about the y-axis

Each pair x and -x has the same y-value, so reflected points match across the y-axis. The graph is symmetric about the y-axis.

13. For y = 12/sqrt(x), what is y at x = 9?

Answer A: 4

sqrt(9) = 3, so y = 12/3 = 4. The coordinate is (9, 4), and x = 9 is in the domain x > 0.

14. What is the real-number domain of y = sqrt(x)?

Answer B: x >= 0

The expression sqrt(x) is real when x is non-negative, including x = 0. Therefore its domain is x >= 0.

15. For y = 3(2^x) + 1, what is y when x = 3?

Answer C: 25

2^3 = 8, so y = 3(8) + 1 = 25. Thus (3, 25) lies on the exponential graph.

16. For y = 7(3^x) + 5, which horizontal line is the graph's asymptote as x decreases without bound?

Answer D: y = 5

Because 3^x approaches 0, the term 7(3^x) approaches 0 and y approaches 5. The horizontal asymptote is y = 5.

17. A quantity is modelled by Q = 160(2^(-t)), where t is measured in hours. What is Q at t = 3 hours?

Answer A: 20

Q = 160(2^(-3)) = 160/8 = 20. Equivalently, three halvings take 160 to 80, then 40, then 20.

18. An exponential graph contains the coordinates (0, 6), (1, 12) and (2, 24). By what factor does y change whenever x increases by 1?

Answer B: It doubles

12/6 = 2 and 24/12 = 2, so each unit increase in x multiplies y by 2. The output doubles.

19. The curve y = x^2 - 1 and the line y = x + 1 intersect exactly at (-1, 0) and (2, 3). What are the graphical solutions of x^2 - 1 = x + 1?

Answer C: x = -1 or x = 2

At both intersections the curve and line have equal y-values. Their x-coordinates are -1 and 2, so these are the two graphical solutions.

20. All the x-axis intersections of a curve are exactly (-3, 0), (1, 0) and (4, 0). Which list gives all roots of the function?

Answer D: -3, 1 and 4

Roots are x-values where y = 0. The three stated x-axis intersections therefore give roots -3, 1 and 4.

21. A tangent to a curve passes through the explicit points (2, 1) and (6, 13). What is the tangent's gradient?

Answer A: 3

The gradient is (13 - 1)/(6 - 2) = 12/4 = 3. This estimates the curve's gradient at the tangent's contact point.

22. At a stated contact point, a curve has tangent y = -2x + 7. What is the curve's estimated gradient there?

Answer B: -2

The tangent's equation has x-coefficient -2, so its gradient is -2. The negative sign means the curve is falling at the contact point.

23. A graph is described as one smooth curve with a single minimum turning point, symmetry about a vertical line and two arms that rise. Which family fits this description?

Answer C: Quadratic

An upward-opening quadratic has one minimum turning point and is symmetric about a vertical line. The other listed families do not have all these features.

24. A table is constructed for y = 2x^3 - x. Which coordinate belongs to x = -2?

Answer D: (-2, -14)

At x = -2, y = 2(-2)^3 - (-2) = 2(-8) + 2 = -14. The coordinate is (-2, -14).