1. **Problem (a):** Sketch the linear function $y = 2x + 3$ and find its slope and y-intercept. 2. The general form of a linear function is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. 3. For $y = 2x + 3$, the slope $m = 2$ and the y-intercept $b = 3$. 4. The slope $2$ means the line rises 2 units vertically for every 1 unit it moves horizontally. 5. The y-intercept $3$ means the line crosses the y-axis at the point $(0,3)$. --- 1. **Problem (b):** Sketch the quadratic function $y = -x^2 + 4x - 3$ and identify the vertex, axis of symmetry, and intercepts. 2. The quadratic function is in the form $y = ax^2 + bx + c$ with $a = -1$, $b = 4$, and $c = -3$. 3. The vertex formula is $x = -\frac{b}{2a} = -\frac{4}{2(-1)} = 2$. 4. Substitute $x=2$ into the function to find the vertex's y-coordinate: $$y = -(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1$$ So, the vertex is at $(2,1)$. 5. The axis of symmetry is the vertical line $x = 2$. 6. To find the y-intercept, set $x=0$: $$y = -0 + 0 - 3 = -3$$ So, the y-intercept is $(0,-3)$. 7. To find the x-intercepts, solve $-x^2 + 4x - 3 = 0$: Multiply both sides by $-1$: $$x^2 - 4x + 3 = 0$$ Factor: $$(x - 3)(x - 1) = 0$$ So, $x = 1$ or $x = 3$. The x-intercepts are $(1,0)$ and $(3,0)$. --- 1. **Problem (c):** Solve graphically the system: $$y = x^2 - 3x + 2$$ and $$y = 2x - 1$$ 2. To find the intersection points, set the two expressions equal: $$x^2 - 3x + 2 = 2x - 1$$ 3. Rearrange to form a quadratic equation: $$x^2 - 3x + 2 - 2x + 1 = 0$$ $$x^2 - 5x + 3 = 0$$ 4. Use the quadratic formula: $$x = \frac{5 \pm \sqrt{(-5)^2 - 4(1)(3)}}{2(1)} = \frac{5 \pm \sqrt{25 - 12}}{2} = \frac{5 \pm \sqrt{13}}{2}$$ 5. Approximate the roots: $$x_1 \approx \frac{5 + 3.606}{2} = 4.303$$ $$x_2 \approx \frac{5 - 3.606}{2} = 0.697$$ 6. Find corresponding y-values using $y = 2x - 1$: $$y_1 = 2(4.303) - 1 = 7.606$$ $$y_2 = 2(0.697) - 1 = 0.394$$ 7. The solutions (intersection points) are approximately $(4.303, 7.606)$ and $(0.697, 0.394)$. --- **Final answers:** - (a) Slope = 2, y-intercept = 3 - (b) Vertex = $(2,1)$, axis of symmetry $x=2$, x-intercepts $(1,0)$ and $(3,0)$, y-intercept $(0,-3)$ - (c) Intersection points approximately $(4.303, 7.606)$ and $(0.697, 0.394)$