Key features of graphs; intersection points via technology practice questions.

A graph has equation y = (x − 3)(x + 5). State the x-coordinates of its x-intercepts.

• x = 3 and x = 5
• x = −3 and x = 5
• x = 3 and x = −5
• x = −3 and x = −5

Answer: x = 3 and x = −5

The x-intercepts (zeros) occur where each factor equals 0: x − 3 = 0 gives x = 3, and x + 5 = 0 gives x = −5.

The graph of y = 3/(x − 2) + 4 has a vertical asymptote and a horizontal asymptote. State their equations.

• x = 2 and y = 4
• x = −2 and y = 4
• x = 4 and y = 2
• x = 2 and y = 0

Answer: x = 2 and y = 4

The function is undefined at x = 2 (vertical asymptote), and as x → ±∞ the fraction → 0 so y → 4 (horizontal asymptote).

Determine the coordinates of the local minimum of the graph of y = x² − 8x + 10.

• (4, −6)
• (−4, 58)
• (4, 10)
• (8, 10)

Answer: (4, −6)

The axis of symmetry is x = −b/(2a) = 8/2 = 4; substituting gives y = 16 − 32 + 10 = −6, so the minimum is (4, −6).

Using technology, the graphs of y = e^x and y = 3 intersect at one point. Determine the x-coordinate to 3 s.f.

• 1.10
• 0.910
• 1.50
• 20.1

Answer: 1.10

Solving e^x = 3 gives x = ln 3 = 1.0986… ≈ 1.10 to 3 s.f.

Determine the y-coordinate of the vertex of the graph y = −(x − 3)² + 7.

• 7
• −7
• 3
• −3

Answer: 7

In vertex form y = a(x − h)² + k the vertex is (h, k) = (3, 7), so the y-coordinate is 7 (a maximum since a < 0).

State where y = 1/(x + 4) is undefined.

• x = 4
• x = −4
• y = 4
• x = 0

Answer: x = −4

The denominator x + 4 = 0 when x = −4, where the function is undefined, giving the vertical asymptote x = −4.