This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Okay, asanda, let's determine the equations for these cubic functions. The general form is y = ax^3 + bx^2 + cx + d. We'll use the x-intercepts and a given point to find the specific equation for each graph. Graph (1) Step 1: Identify x-intercepts and their multiplicity. The graph touches the x-axis at x=-1, indicating a double root (multiplicity 2). It crosses the x-axis at x=3, indicating a single root. The factored form is y = a(x+1)^2(x-3). Step 2: Use the given point (2, -6) to find a. Substitute x=2 and y=-6 into the equation: -6 = a(2+1)^2(2-3) -6 = a(3)^2(-1) -6 = a(9)(-1) -6 = -9a a = (-6)/(-9) = (2)/(3) Step 3: Write the equation in factored form and expand. y = (2)/(3)(x+1)^2(x-3) Expand (x+1)^2: (x+1)^2 = x^2 + 2x + 1 Multiply by (x-3): (x^2 + 2x + 1)(x-3) = x^3 - 3x^2 + 2x^2 - 6x + x - 3 = x^3 - x^2 - 5x - 3 Multiply by (2)/(3): y = (2)/(3)(x^3 - x^2 - 5x - 3) y = (2)/(3)x^3 - (2)/(3)x^2 - (10)/(3)x - 2 Graph (2) Step 1: Identify x-intercepts and their multiplicity. The graph crosses the x-axis at x=-2, indicating a single root. It touches the x-axis at x=3, indicating a double root. The factored form is y = a(x+2)(x-3)^2. Step 2: Use the given point (0, -18) to find a. Substitute x=0 and y=-18 into the equation: -18 = a(0+2)(0-3)^2 -18 = a(2)(-3)^2 -18 = a(2)(9) -18 = 18a a = (-18)/(18) = -1 Step 3: Write the equation in factored form and expand. y = -1(x+2)(x-3)^2 Expand (x-3)^2: (x-3)^2 = x^2 - 6x + 9 Multiply by (x+2): (x+2)(x^2 - 6x + 9) = x^3 - 6x^2 + 9x + 2x^2 - 12x + 18 = x^3 - 4x^2 - 3x + 18 Multiply by -1: y = -(x^3 - 4x^2 - 3x + 18) y = -x^3 + 4x^2 + 3x - 18 Graph (3) Step 1: Identify x-intercepts and their multiplicity. The graph crosses the x-axis at x=-4, indicating a single root. It touches the x-axis at x=0, indicating a double root. The factored form is y = a(x+4)(x-0)^2, which simplifies to y = ax^2(x+4). Step 2: Use the given point (-2, 4) to find a. Substitute x=-2 and y=4 into the equation: 4 = a(-2)^2(-2+4) 4 = a(4)(2) 4 = 8a a = (4)/(8) = (1)/(2) Step 3: Write the equation in factored form and expand. y = (1)/(2)x^2(x+4) y = (1)/(2)(x^3 + 4x^2) y = (1)/(2)x^3 + 2x^2 Graph (4) Step 1: Identify x-intercepts and their multiplicity. The graph crosses the x-axis at x=0, indicating a single root. It touches the x-axis at x=5, indicating a double root. The factored form is y = a(x-0)(x-5)^2, which simplifies to y = ax(x-5)^2. Step 2: Use the given point (1, -4) to find a. Substitute x=1 and y=-4 into the equation: -4 = a(1)(1-5)^2 -4 = a(1)(-4)^2 -4 = a(1)(16) -4 = 16a a = (-4)/(16) = -(1)/(4) Step 3: Write the equation in factored form and expand. y = -(1)/(4)x(x-5)^2 Expand (x-5)^2: (x-5)^2 = x^2 - 10x + 25 Multiply by x: x(x^2 - 10x + 25) = x^3 - 10x^2 + 25x Multiply by -(1)/(4): y = -(1)/(4)(x^3 - 10x^2 + 25x) y = -(1)/(4)x^3 + (5)/(2)x^2 - (25)/(4)x 3 done, 2 left today. You're making progress.