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What is the \( y \)-intercept of the following function? \( f(x)=(x+1)(x-4)(x+3)(x-1) \) 9 4 2 12

Ask by Donnelly Haynes. in the United States
Jan 30,2025

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The \( y \)-intercept is 12.

Solution

To find the \( y \)-intercept of the function \( f(x) = (x+1)(x-4)(x+3)(x-1) \), we evaluate the function at \( x = 0 \). \[ f(0) = (0 + 1)(0 - 4)(0 + 3)(0 - 1) \] Calculate each term step-by-step: 1. \( 0 + 1 = 1 \) 2. \( 0 - 4 = -4 \) 3. \( 0 + 3 = 3 \) 4. \( 0 - 1 = -1 \) Now multiply these results: \[ f(0) = 1 \times (-4) \times 3 \times (-1) \] \[ f(0) = (-4) \times 3 = -12 \] \[ -12 \times (-1) = 12 \] So, the \( y \)-intercept is **12**. **Answer:** 12

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Bonus Knowledge

To find the \( y \)-intercept of the function \( f(x) = (x+1)(x-4)(x+3)(x-1) \), you can evaluate \( f(0) \). This means substituting \( x = 0 \) into the function. Calculating it gives: \[ f(0) = (0+1)(0-4)(0+3)(0-1) = 1 \cdot (-4) \cdot 3 \cdot (-1) = 12. \] So, the \( y \)-intercept is \( 12 \). Now, let’s keep this math party going! To remember how to graph these functions later, always think of the \( y \)-intercept as where the graph meets the \( y \)-axis. It's fun to visualize how even polynomial functions can have shapes that twist and turn around those intercepts! If you’re curious about exploring polynomial functions further, consider reading up on polynomial long division or synthetic division. These techniques can help when you need to factor polynomials or find roots, and they can open the door to deeper algebraic adventures!

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