A rocket's path is modeled by the equation y=−2x^2 +4x+6‍ . Graph the function. Identify the x-intercepts and interpret their meaning in the context of the rocket launch.(2 points) x-intercepts (list smaller one first): and Question 2 Using the context, interpret the meaning of the above x-intercept(s).(2 points)

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GPT-4o mini · Answer

To analyze the function $ y = -2x^2 + 4x + 6 $, let's first find the x-intercepts by setting $ y = 0 $: $$ 0 = -2x^2 + 4x + 6 $$ We can rearrange the equation: $$ -2x^2 + 4x + 6 = 0 $$ Divide the entire equation by -2 to simplify: $$ x^2 - 2x - 3 = 0 $$ Next, we factor the quadratic: $$ (x - 3)(x + 1) = 0 $$ Setting each factor to zero gives the x-intercepts: $$ x - 3 = 0 \quad \Rightarrow \quad x = 3 $$ $$ x + 1 = 0 \quad \Rightarrow \quad x = -1 $$ Thus, the x-intercepts are $ x = -1 $ and $ x = 3 $. ### Question 1: X-intercepts **X-Intercepts:** - Smaller one first: $ -1, 3 $ ### Question 2: Interpretation of the X-Intercepts The x-intercepts represent the points in time (in whatever time unit is chosen) when the rocket has a height of $ y = 0 $, meaning it is at ground level. 1. **X-Intercept $ x = -1 $**: This intercept occurs before the launch at $ x = -1 $. In the context of the rocket launch, this could represent a theoretical scenario or extension of the model where the rocket has not yet been launched (for instance, if time is represented in seconds, this could correlate to a situation that doesn't correspond to a real launch). 2. **X-Intercept $ x = 3 $**: This intercept represents the rocket returning to the ground at $ x = 3 $ seconds after launch. This means the rocket reaches its maximum height and then descends back to ground level approximately 3 seconds after being launched. Overall, the x-intercepts inform us not only about the specific moments when the rocket is at ground level but also provide insight into the duration of the rocket's flight.