Quadratic Properties 9Ec57B

1. **Problem 1:** Given $f(x) = -3x^2 + 9x + 54$, find the x-intercepts, y-intercept, vertex, and line of symmetry. 2. **Find x-intercepts:** Set $f(x) = 0$: $$-3x^2 + 9x + 54 = 0$$ Divide both sides by $-3$: $$\cancel{-3}x^2 + \cancel{-3}(-3x) + \cancel{-3}(-18) = 0 \Rightarrow x^2 - 3x - 18 = 0$$ Use quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=-3$, $c=-18$: $$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-18)}}{2(1)} = \frac{3 \pm \sqrt{9 + 72}}{2} = \frac{3 \pm \sqrt{81}}{2} = \frac{3 \pm 9}{2}$$ So, $$x_1 = \frac{3 - 9}{2} = \frac{-6}{2} = -3$$ $$x_2 = \frac{3 + 9}{2} = \frac{12}{2} = 6$$ The smallest x-intercept is $-3$ and the largest is $6$. 3. **Find y-intercept:** Set $x=0$: $$f(0) = -3(0)^2 + 9(0) + 54 = 54$$ So, y-intercept is $54$. 4. **Find vertex:** Vertex formula $x = -\frac{b}{2a}$ for $f(x) = ax^2 + bx + c$ with $a = -3$, $b=9$: $$x = -\frac{9}{2(-3)} = -\frac{9}{-6} = \frac{3}{2} = 1.5$$ Find $y$ at $x=1.5$: $$f(1.5) = -3(1.5)^2 + 9(1.5) + 54 = -3(2.25) + 13.5 + 54 = -6.75 + 13.5 + 54 = 60.75$$ Vertex is $(1.5, 60.75)$. 5. **Line of symmetry:** Vertical line through vertex $x=1.5$. --- 6. **Problem 2:** Given $f(x) = x^2 - 4x - 21$, find the vertex. Use vertex formula $x = -\frac{b}{2a}$ with $a=1$, $b=-4$: $$x = -\frac{-4}{2(1)} = \frac{4}{2} = 2$$ Find $y$ at $x=2$: $$f(2) = (2)^2 - 4(2) - 21 = 4 - 8 - 21 = -25$$ Vertex is $(2, -25)$. --- 7. **Problem 3:** Given $y = 2x^2 + 12x + 9$, find vertex and determine if max or min. Vertex $x = -\frac{b}{2a} = -\frac{12}{2(2)} = -\frac{12}{4} = -3$ Find $y$ at $x=-3$: $$y = 2(-3)^2 + 12(-3) + 9 = 2(9) - 36 + 9 = 18 - 36 + 9 = -9$$ Since $a=2 > 0$, parabola opens upward, so vertex is a minimum. **Answer:** Vertex is $(-3, -9)$; minimum. --- **Summary:** - Problem 1: - Smallest x-intercept: $-3$ - Largest x-intercept: $6$ - y-intercept: $54$ - Vertex: $(1.5, 60.75)$ - Line of symmetry: $x=1.5$ - Problem 2: - Vertex: $(2, -25)$ - Problem 3: - Vertex: $(-3, -9)$ - Minimum