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Finding the Intercepts of
To find the intercepts of the parabola :
Finding the y-intercept: Let and substitute it into the equation to solve for :
The y-intercept is the point .
Finding the x-intercepts: Let and solve for :
Factor the quadratic equation:
Use the zero product property to set each factor equal to zero:
Solve the equations:
The x-intercepts are the points and .
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Ch.10 Quadratic Equations - Elementary Algebra @ OpenStax
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Finding the Intercepts of
Finding the Intercepts of
A logistics manager uses a quadratic function to model the fuel efficiency of a delivery fleet based on speed. To find the 'zero-efficiency' points where the graph crosses the horizontal axis (the x-intercepts), which algebraic step is required?
A landscape architect uses a quadratic equation to model the path of water from a fountain. Match the type of intercept the architect needs to find with the correct algebraic procedure.
A small business owner uses a quadratic equation to model the daily operating costs of a new storefront. To find the y-intercept, which represents the fixed costs on day zero, the owner should substitute 0 for y and solve the equation for x.
A marketing specialist is analyzing a campaign's effectiveness using a parabolic model . To find the points where the effectiveness is zero (the x-intercepts), the specialist follows a standard algebraic procedure. Arrange the following steps in the correct order.
Calculating X-Intercepts for a Parabolic Model
A small business owner uses a quadratic model to predict the monthly profit () based on the number of years in operation (). To find the 'starting profit' on the day the business opened (the y-intercept), the owner must substitute the value ____ for the variable .
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A structural engineer is analyzing a parabolic bridge arch modeled by the equation . When identifying the x-intercepts (the points where the arch meets the horizontal foundation), which coordinate must always be equal to zero at those points?
A human resources manager uses a parabolic function to model the number of employee departures at a new branch over time. If the manager determines that the y-intercept of this model is 12, which ordered pair correctly identifies this point on a coordinate graph?
Learn After
A startup company uses the quadratic function y = x^2 + 4x + 3 to model their projected monthly revenue. In this model, the y-intercept represents the revenue at the very beginning of the project (month zero). What is the y-intercept for this function?
A structural engineer is using the quadratic equation y = x^2 + 4x + 3 to model the curve of a bridge support. The engineer calculates that the x-intercepts, which represent the points where the support meets the horizontal base (y = 0), are located at the coordinates (-1, 0) and (-3, 0). Is this calculation correct?
A project manager uses the equation y = x^2 + 4x + 3 to model the variance in project costs over time. Arrange the following steps in the correct order to solve for the x-intercepts (the points where variance is zero) for this specific model.
A landscape architect is using the quadratic equation to model the elevation of a small pedestrian tunnel, where is the height in feet and is the horizontal distance. Match each intercept description with its correct coordinate point.
Zero-Variance Baseline Identification
A warehouse manager uses the formula to model the fluctuation in storage space requirements throughout a production cycle. The x-intercepts of this formula represent the points where the warehouse is at its baseline capacity (). After factoring the formula into $0 = (x+1)(x+3)x = -1$. The x-coordinate of the other baseline point is ____.
Technical Documentation: Intercepts of the Model
Logistics Budget Variance Analysis
An operations analyst uses the quadratic equation to model the deviation in a supply chain process. To find the points where the deviation is zero (the x-intercepts), the analyst must identify the factored form of the expression. According to the standard solution for this equation, which of the following is the correct factored form of ?
A software engineer is developing a calculation module for a business model represented by the equation . To find the -intercepts of the model, the engineer factors the equation to get $0 = (x + 1)(x + 3)x$-intercepts?