Plot the graph of f (x) = x 2 − x − 6, − 3 ≤ x ≤ 4, x ∈ . (a) Use the graph to solve f (x) = 0 and to fi nd the y-intercept. Verify the results algebraically. (b) Use the graph to solve f (x) = 2. Verify the results algebraically. Give all the answers correct to one decimal place.
Solution x –3 –2 –1 0 1 2 3 4 f
y
3 4 5 6
y = 2 –4
–2·4 –3
1 2
3·4 –2 –1 0
–1 –2 –3 –4 –5 –6
1 2 3 4 x f (x) 6 0 –4 –6 –6 –4 0 6
(a) Graph: y = 0: x = −2, 3
[The roots are where the graph cuts the x-axis.]
y-intercept: y = − 6 Algebra: f (x) = 0: x 2 − x − 6 = 0 (x + 2)(x − 3) = 0 x = −2, 3 y-intercept: f (0) = (0) 2 − (0) − 6 = − 6
(b) Draw the line y = 2.
Graph: f (x) = 2: x = −2·4, 3·4
Algebra: f (x) = 2: x 2 − x − 6 = 2 x 2 − x − 8 = 0
Use the quadratic formula: a = 1, b = –1, c = –8 x =
