To help you understand how to graph the function \( y = -x^2 - 80x + 1200 \), I can guide you through the steps. However, I cannot generate graphical images directly.
Step 1: Identify the Type of Function
The given function is a quadratic function of the form \( y = ax^2 + bx + c \).
- Here, \( a = -1 \), \( b = -80 \), and \( c = 1200 \).
- Since \( a < 0 \), this parabola opens downward.
Step 2: Find the Vertex
The vertex of a quadratic function can be found using the formula: \[ x = -\frac{b}{2a} \] Substituting the values of \( a \) and \( b \): \[ x = -\frac{-80}{2(-1)} = \frac{80}{-2} = -40 \]
Now substitute \( x = -40 \) back into the function to find \( y \): \[ y = -(-40)^2 - 80(-40) + 1200 \] \[ y = -1600 + 3200 + 1200 \] \[ y = 2800 \]
So the vertex is at \( (-40, 2800) \).
Step 3: Find the Y-Intercept
The y-intercept occurs when \( x = 0 \): \[ y = -0^2 - 80 \cdot 0 + 1200 = 1200 \] So the y-intercept is \( (0, 1200) \).
Step 4: Find the X-Intercepts
To find the x-intercepts, set \( y = 0 \): \[ 0 = -x^2 - 80x + 1200 \] This can be rewritten as: \[ x^2 + 80x - 1200 = 0 \]
Next, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ a = 1, b = 80, c = -1200 \] Substituting into the formula: \[ x = \frac{-80 \pm \sqrt{80^2 - 4 \cdot 1 \cdot (-1200)}}{2 \cdot 1} \] \[ x = \frac{-80 \pm \sqrt{6400 + 4800}}{2} \] \[ x = \frac{-80 \pm \sqrt{11200}}{2} \]
Calculate \( \sqrt{11200} \approx 105.83 \): \[ x = \frac{-80 \pm 105.83}{2} \]
So the roots are:
- \( x_1 = \frac{25.83}{2} \approx 12.915 \)
- \( x_2 = \frac{-185.83}{2} \approx -92.915 \)
Step 5: Graphing
- Plot the vertex \( (-40, 2800) \).
- Plot the y-intercept \( (0, 1200) \).
- Plot the x-intercepts \( (12.915, 0) \) and \( (-92.915, 0) \).
- Sketch the parabola opening downward, passing through these points.
Now that you have the critical points and the shape of the graph, you can use graphing tools or software to create the graph based on this information! If you are using a graphing calculator or online graphing tool, input \( y = -x^2 - 80x + 1200 \) to visualize it directly.