Answer: 43) y = -x² - 2x + 3
44) y = (1/2)x² + 2x + 3
Step-by-step explanation:
43) Since the x-intercepts and another point are given, use the Intercept form:
y = a(x - p)(x - q) where
- p and q are the x-intercepts
- "a" is the vertical stretch
- (x, y) is another point on the curve
Given: p = -3, q = 1 (x, y) = (-1, 4)
y = a(x - p)(x - q)
y = a(x + 3)(x - 1)
4 = a(-1 + 3)(-1 - 1)
4 = a(2)(-2)
4 = -4a
-1 = a
Equation: y = -1(x + 3)(x - 1)
Expand: y = -(x² - x + 3x - 3)
= -(x² + 2x - 3)
= -x² - 2x + 3
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Since the vertex and another point are given, use the Vertex form:
y = a(x - h)² + k where
- (h, k) is the vertex
- "a" is the vertical stretch
- (x, y) is another point on the curve
Given: (h, k) = (-2, 1) and (x, y) = (0, 3)
y = a(x - h)² + k
y = a(x + 2)² + 1
3 = a(0 + 2)² + 1
3 = 4a + 1
2 = 4a
1/2 = a
Equation: y = (1/2)(x + 2)² + 1
Expand: y = (1/2)(x² + 4x + 4) + 1
= (1/2)x² + 2x + 2 + 1
= (1/2)x² + 2x + 3
