Learn Before
Finding the Intercepts of
To find the intercepts of the parabola :
Finding the y-intercept: Substitute into the equation and compute :
The y-intercept is the point .
Finding the x-intercepts: Substitute into the equation and solve for :
Factor out to simplify the quadratic expression:
Recognize and factor the perfect square trinomial:
Apply the Zero Product Property:
Since there is only one real solution, the parabola has exactly one x-intercept at the point .
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Ch.10 Quadratic Equations - Elementary Algebra @ OpenStax
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Finding the Intercepts of
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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 .
Analyzing Startup Valuation Trends
Documenting Procedures for Energy Efficiency Modeling
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 production manager uses the quadratic equation y = -x^2 - 12x - 36 to model the cost deviation of a manufacturing process. Which of the following points represents the y-intercept of this model?
True or False: The quadratic function y = -x^2 - 12x - 36, which can be used to model a company's production cost deviation, has exactly one x-intercept at the point (-6, 0).
A quality control technician uses the quadratic model to analyze the deviation in a production process. To find the x-intercept (the point where the deviation is zero), the technician must follow a specific algebraic sequence. Arrange the following steps in the correct order as they would appear in the solution process.
An operations manager uses the quadratic function to model the deviation in production costs. Match each component of the model's intercepts with its correct numerical value.
Identifying Zero-Deviation Points in Quality Control
Inventory Deviation Analysis
An industrial engineer uses the model to represent the deviation in a conveyor belt's speed. After factoring the equation into $0 = -(x + 6)^2to find the x-intercept, the engineer applies the ____ Property to solve for the value ofx$.
Documentation of Intercept Analysis for Performance Models
A technician uses the quadratic function to model the cost deviation of a production line. To find the point of zero deviation (the x-intercept), the technician must factor the equation. According to the solution process for this specific model, which of the following is the correct factored form of the equation?
When following the solution process to find the x-intercepts of the performance model , the expression is identified as which specific type of algebraic structure?