1. **Stating the problem:** We are given the quadratic function $$f(x) = x^2 - 14x + 48$$ and need to describe its graph by identifying key features such as intercepts, direction of opening, and axis of symmetry. 2. **Formula and rules:** - The quadratic function is in standard form $$ax^2 + bx + c$$ where $$a=1$$, $$b=-14$$, and $$c=48$$. - The parabola opens upward if $$a > 0$$ and downward if $$a < 0$$. - The axis of symmetry is given by $$x = -\frac{b}{2a}$$. - The y-intercept is the value of $$f(0) = c$$. - The x-intercepts (roots) are found by solving $$f(x) = 0$$. 3. **Finding the y-intercept:** - Substitute $$x=0$$ into the function: $$f(0) = 0^2 - 14 \times 0 + 48 = 48$$ - So, the y-intercept is at $$(0, 48)$$. 4. **Direction of the parabola:** - Since $$a=1 > 0$$, the parabola opens **upward**. 5. **Finding the x-intercepts:** - Solve $$x^2 - 14x + 48 = 0$$. - Factor the quadratic: $$x^2 - 14x + 48 = (x - 6)(x - 8) = 0$$ - So, the roots are $$x=6$$ and $$x=8$$. - The x-intercepts are $$(6, 0)$$ and $$(8, 0)$$. 6. **Finding the axis of symmetry:** - Use formula: $$x = -\frac{b}{2a} = -\frac{-14}{2 \times 1} = \frac{14}{2} = 7$$ - The axis of symmetry is $$x=7$$. **Summary:** - The y-intercept is $$(0, 48)$$. - The parabola opens **upward**. - The x-intercepts are $$(6, 0)$$ and $$(8, 0)$$. - The axis of symmetry is $$x=7$$. **Note:** The statement "The parabola opens downward" and "The axis of symmetry is $$x=6$$" are incorrect based on the calculations. **Final answer:** - The y-intercept is $$(0, 48)$$. - The parabola opens upward. - The x-intercepts are $$(6, 0)$$ and $$(8, 0)$$. - The axis of symmetry is $$x=7$$.