1. **State the problem:** Given the quadratic function $y = 4x^2 - 10x + 8$, analyze its key features including intercepts, vertex, domain, range, and behavior. 2. **Formula and rules:** The quadratic function is in standard form $y = ax^2 + bx + c$ where $a=4$, $b=-10$, and $c=8$. 3. **Find x-intercepts:** Solve $4x^2 - 10x + 8 = 0$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 4 \cdot 8}}{2 \cdot 4} = \frac{10 \pm \sqrt{100 - 128}}{8}$$ Since $100 - 128 = -28 < 0$, no real roots exist, so no x-intercepts. (Note: The user’s given x-intercepts (1,0) and (2,0) are incorrect for this function.) 4. **Find y-intercept:** Set $x=0$: $$y = 4(0)^2 - 10(0) + 8 = 8$$ So y-intercept is $(0,8)$. 5. **Find vertex:** Use vertex formula $x = -\frac{b}{2a} = -\frac{-10}{2 \cdot 4} = \frac{10}{8} = \frac{5}{4}$. Calculate $y$ at $x=\frac{5}{4}$: $$y = 4\left(\frac{5}{4}\right)^2 - 10\left(\frac{5}{4}\right) + 8 = 4 \cdot \frac{25}{16} - \frac{50}{4} + 8 = \frac{100}{16} - \frac{200}{16} + \frac{128}{16} = \frac{28}{16} = \frac{7}{4} = 1.75$$ So vertex is $\left(\frac{5}{4}, \frac{7}{4}\right)$. 6. **Domain:** All real numbers $\mathbb{R}$. 7. **Range:** Since $a=4 > 0$, parabola opens upward, so minimum value at vertex $y = \frac{7}{4}$, thus range is $y \geq \frac{7}{4}$. 8. **Increasing/decreasing:** Decreasing on $(-\infty, \frac{5}{4})$, increasing on $(\frac{5}{4}, \infty)$. 9. **Axis of symmetry:** Vertical line $x = \frac{5}{4}$. 10. **End behavior:** As $x \to \pm \infty$, $y \to \infty$. **Final corrected key points:** - No real x-intercepts - y-intercept at $(0,8)$ - Vertex at $\left(\frac{5}{4}, \frac{7}{4}\right)$ - Domain $\mathbb{R}$ - Range $y \geq \frac{7}{4}$ - Parabola opens upward - Axis of symmetry $x=\frac{5}{4}$ - Decreasing on $x < \frac{5}{4}$, increasing on $x > \frac{5}{4}$