1. **Problem 5a:** Given the function $y = x \cdot \left(x - \frac{3}{4}\right)$, find the zeros, vertex, stretch factor, general form, and y-intercept. 2. **Formula and rules:** - To find zeros, set $y=0$ and solve for $x$. - The vertex of a parabola in factored form $y = a(x - x_1)(x - x_2)$ is at $x = \frac{x_1 + x_2}{2}$. - The stretch factor is the coefficient $a$. - The general form is $y = ax^2 + bx + c$. - The y-intercept is found by evaluating $y$ at $x=0$. 3. **Find zeros:** $$0 = x \cdot \left(x - \frac{3}{4}\right)$$ Zeros are $x=0$ and $x=\frac{3}{4}$. 4. **Find vertex:** $$x_{vertex} = \frac{0 + \frac{3}{4}}{2} = \frac{3}{8}$$ Evaluate $y$ at $x=\frac{3}{8}$: $$y = \frac{3}{8} \cdot \left(\frac{3}{8} - \frac{3}{4}\right) = \frac{3}{8} \cdot \left(-\frac{3}{8}\right) = -\frac{9}{64}$$ Vertex is $\left(\frac{3}{8}, -\frac{9}{64}\right)$. 5. **Stretch factor:** Coefficient $a=1$. 6. **General form:** Expand: $$y = x^2 - \frac{3}{4}x$$ 7. **Y-intercept:** At $x=0$, $$y = 0$$ --- 1. **Problem 6a:** Show that the three forms represent the same function for the first example: - Vertex form: $y = (x - 0.5)^2 - 2.25$ - General form: $y = x^2 - x - 2$ - Factored form: $y = (x - 2)(x + 1)$ 2. **Expand vertex form:** $$y = (x - 0.5)^2 - 2.25 = x^2 - 2 \cdot 0.5 x + 0.5^2 - 2.25 = x^2 - x + 0.25 - 2.25 = x^2 - x - 2$$ Matches general form. 3. **Expand factored form:** $$y = (x - 2)(x + 1) = x^2 + x - 2x - 2 = x^2 - x - 2$$ Matches general form. Thus, all three forms represent the same function. --- 1. **Problem 7a:** Find equations of parabolas with zeros $x_1=1$ and $x_2=3$. 2. **General form from zeros:** $$f(x) = a(x - x_1)(x - x_2) = a(x - 1)(x - 3)$$ 3. **Choose $a=1$ for simplicity:** $$f(x) = (x - 1)(x - 3) = x^2 - 4x + 3$$ 4. **Alternative parabola:** Choose $a = -1$: $$f(x) = - (x - 1)(x - 3) = -x^2 + 4x - 3$$ These are two parabolas with the given zeros. --- **Final answers:** **5a:** Zeros: $0$, $\frac{3}{4}$; Vertex: $\left(\frac{3}{8}, -\frac{9}{64}\right)$; Stretch factor: $1$; General form: $y = x^2 - \frac{3}{4}x$; Y-intercept: $0$. **6a:** The three forms represent the same function $y = x^2 - x - 2$. **7a:** Parabolas with zeros $1$ and $3$ are $f(x) = (x - 1)(x - 3)$ and $f(x) = - (x - 1)(x - 3)$.