4. Brandon is writing an equation for graph B. He says he can determine the equation using the solutions and the y-intercept. Do you agree or disagree? Explain your answer.

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UpStudy AI · Accepted Answer

Yes, I agree with Brandon. Knowing the solutions (which are the x-intercepts) and the y-intercept provides enough information to determine the equation of the graph, especially if the graph represents a linear or quadratic function. Here's why:

1. **Linear Functions (Straight Lines):**
   - **Solutions (x-intercepts):** For a linear equation $$ y = mx + b $$, knowing one x-intercept allows you to find the slope $$ m $$ if you also know the y-intercept $$ b $$.
   - **Y-intercept ($$ b $$):** This directly gives you the value where the line crosses the y-axis.
   - **Example:** If the solutions are $$ x = 2 $$ and the y-intercept is $$ (0, -4) $$, the equation can be determined as follows:
     $$
     y = m(x - 2)
     $$
     Since $$ y = -4 $$ when $$ x = 0 $$:
     $$
     -4 = m(0 - 2) \implies m = 2
     $$
     So, the equation is $$ y = 2(x - 2) $$ or $$ y = 2x - 4 $$.

2. **Quadratic Functions (Parabolas):**
   - **Solutions (x-intercepts):** For a quadratic equation $$ y = a(x - r_1)(x - r_2) $$, knowing the two roots $$ r_1 $$ and $$ r_2 $$ allows you to form the basic structure of the equation.
   - **Y-intercept:** This provides the value of $$ y $$ when $$ x = 0 $$, which can be used to solve for the coefficient $$ a $$.
   - **Example:** If the solutions are $$ x = 1 $$ and $$ x = 3 $$, and the y-intercept is $$ (0, -6) $$:
     $$
     y = a(x - 1)(x - 3)
     $$
     Plugging in the y-intercept:
     $$
     -6 = a(0 - 1)(0 - 3) \implies -6 = a(3) \implies a = -2
     $$
     So, the equation is $$ y = -2(x - 1)(x - 3) $$ or $$ y = -2x^2 + 8x - 6 $$.

3. **Higher-Degree Polynomials:**
   - For polynomials of degree three or higher, simply knowing the solutions and the y-intercept might not be sufficient to determine the equation uniquely, as there could be multiple polynomials of the same degree that share those characteristics. Additional information, such as another point on the graph or specific behavior (like turning points), would be necessary.

**Conclusion:**
If Graph B represents a linear or quadratic function, Brandon can indeed determine its equation using the solutions and the y-intercept. However, for more complex graphs, additional information would be required.
Yes, Brandon can determine the equation of Graph B using the solutions and the y-intercept, especially if the graph is linear or quadratic.

Brandon is writing an equation for graph B.
He says he can determine the equation using
the solutions and the y-intercept. Do you agree
or disagree? Explain your answer.

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