1. **Stating the problem:** We are given two quadratic functions and asked to analyze their properties including vertex, optimal value, axis of symmetry, zeros, y-intercept, direction of opening, step pattern, and equation. 2. **Given functions:** (a) $$y = 3(x - 4)^2 + 8$$ expanded to $$y = 3x^2 - 24x + 56$$ (b) $$y = -1(x^2 + 14x + 49) - 3$$ expanded to $$y = -x^2 - 14x - 52$$ 3. **Analyzing function (a):** - Vertex form is $$y = 3(x - 4)^2 + 8$$ so vertex is at $$ (4, 8) $$. - Since the coefficient of $$ (x-4)^2 $$ is positive (3), the parabola opens upward. - Axis of symmetry is the vertical line $$ x = 4 $$. - Optimal value is the minimum value of $$ y $$, which is the vertex's y-coordinate: $$ 8 $$. - To find zeros, solve $$ 3(x - 4)^2 + 8 = 0 $$: $$ 3(x - 4)^2 = -8 $$ which has no real solutions since the left side is always non-negative and right side is negative. - Y-intercept is found by substituting $$ x=0 $$: $$ y = 3(0 - 4)^2 + 8 = 3(16) + 8 = 48 + 8 = 56 $$. - Step pattern for coefficient 3 means from vertex, steps go: right 1, up 3; right 2, up 12; right 3, up 27. 4. **Analyzing function (b):** - Vertex form is $$ y = -1(x + 7)^2 - 3 $$ (since $$ x^2 + 14x + 49 = (x+7)^2 $$), so vertex is at $$ (-7, -3) $$. - Coefficient of $$ (x+7)^2 $$ is negative (-1), so parabola opens downward. - Axis of symmetry is $$ x = -7 $$. - Optimal value is the maximum value of $$ y $$, which is $$ -3 $$. - To find zeros, solve $$ -1(x + 7)^2 - 3 = 0 $$: $$ -1(x + 7)^2 = 3 $$ $$ (x + 7)^2 = -3 $$ no real solutions. - Y-intercept by substituting $$ x=0 $$: $$ y = -1(0 + 7)^2 - 3 = -49 - 3 = -52 $$. - Step pattern for coefficient -1 means from vertex, steps go: right 1, down 1; right 2, down 4; right 3, down 9. 5. **Summary table for each parabola:** | Property | Parabola (a) | Parabola (b) | |--------------------|------------------------------|------------------------------| | Vertex | (4, 8) | (-7, -3) | | Optimal Value | Minimum = 8 | Maximum = -3 | | Axis of Symmetry | $$ x = 4 $$ | $$ x = -7 $$ | | Zeros | None (no real roots) | None (no real roots) | | Y-intercept | 56 | -52 | | Direction of Opening| Upward | Downward | | Step Pattern | Right 1, Up 3; Right 2, Up 12; Right 3, Up 27 | Right 1, Down 1; Right 2, Down 4; Right 3, Down 9 | | Equation | $$ y = 3(x - 4)^2 + 8 $$ | $$ y = -1(x + 7)^2 - 3 $$ |