15) Tell whether each function is linear. Explain. $$ \begin{array}{ll} ext { a) } 2 y=-3 x^{2} & ext { b) } y=4 x-7\end{array} $$

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15) Tell whether each function is linear. Explain. $$ \begin{array}{ll}	ext { a) } 2 y=-3 x^{2} & 	ext { b) } y=4 x-7\end{array} $$

UpStudy AI · Accepted Answer

Let's determine whether each given function is linear by analyzing their forms.

### a) $$ 2y = -3x^{2} $$

**Rearranged Form:**
$$
y = \frac{-3}{2}x^{2}
$$

**Analysis:**
- **Degree of the Function:** The highest power of $$ x $$ is 2, which makes it a **quadratic** function.
- **Standard Linear Form:** A linear function has the form $$ y = mx + b $$, where the highest power of $$ x $$ is 1.

**Conclusion:**  
This function is **not linear** because the presence of $$ x^{2} $$ makes it a quadratic function, not a straight line.

---

### b) $$ y = 4x - 7 $$

**Form:**
$$
y = 4x - 7
$$

**Analysis:**
- **Degree of the Function:** The highest power of $$ x $$ is 1.
- **Standard Linear Form:** It matches the linear form $$ y = mx + b $$, where:
  - $$ m = 4 $$ is the slope.
  - $$ b = -7 $$ is the y-intercept.

**Conclusion:**  
This function is **linear** because it fits the standard form of a linear equation, representing a straight line when graphed.

---

**Summary:**
- **a)** Not linear (it's a quadratic function due to the $$ x^{2} $$ term).
- **b)** Linear (fits the form $$ y = mx + b $$).
a) Not linear (quadratic function). b) Linear.

Tell whether each function is linear. Explain.

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