Key features of graphs: zeros, intercepts, symmetry, asymptotes practice questions.

Determine the equation of the horizontal asymptote of the graph of f(x) = (3x − 5)/(x + 2).

• y = −2
• y = 3
• x = 3
• y = −5/2

Answer: y = 3

As x → ±∞ the function approaches the ratio of the leading coefficients, 3/1, so the horizontal asymptote is y = 3; x = −2 gives the vertical asymptote instead.

Determine the zeros of the function f(x) = x² − 5x + 6.

• x = −2 and x = −3
• x = 1 and x = 6
• x = 2 and x = 3
• x = −1 and x = −6

Answer: x = 2 and x = 3

Factorising gives f(x) = (x − 2)(x − 3), so the zeros are x = 2 and x = 3. Taking the factors with the wrong sign gives −2 and −3.

Determine the coordinates of the y-intercept of the graph of f(x) = (x − 4)(x + 2).

• (0, −8)
• (0, 8)
• (−8, 0)
• (0, −2)

Answer: (0, −8)

Set x = 0: f(0) = (−4)(2) = −8, giving the point (0, −8). Writing (−8, 0) confuses the y-intercept with an x-intercept.

Determine the equation of the axis of symmetry of the graph of y = x² − 6x + 11.

• x = −3
• x = 6
• y = 3
• x = 3

Answer: x = 3

The axis of symmetry is x = −b/(2a) = 6/2 = 3. Sign errors give x = −3, and it must be a vertical line x = 3, not y = 3.

Determine how many x-intercepts the graph of y = x² + 4x + 5 has.

• 0
• 1
• 2
• It cannot be determined without technology

Answer: 0

Completing the square gives y = (x + 2)² + 1 ≥ 1 > 0 for all x, so the graph never meets the x-axis: there are no x-intercepts.

Determine which of the following graphs is symmetric about the y-axis.

• y = x³
• y = (x − 2)²
• y = x² + 2
• y = 1/x

Answer: y = x² + 2

Replacing x by −x leaves x² + 2 unchanged, so its graph is symmetric about the y-axis. The graph of (x − 2)² is symmetric about x = 2, and x³ and 1/x have point symmetry about the origin.