Quadratic function graphs represent equations of the form y = ax^2 + bx + c, showcasing parabolas in physics, engineering, and economics. These graphs feature a vertex, axis of symmetry, and intercepts, with the coefficient 'a' determining their orientation and width. Understanding their structure and variations is crucial for applications in real-world scenarios, such as modeling forces, projectile trajectories, and financial analyses.
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Quadratic function graphs are visual representations of quadratic equations in the form \(y = ax^2 + bx + c\) depicted as parabolas
Vertex
The vertex is the highest or lowest point on the graph and is determined by the direction of opening and the coefficient \(a\)
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two congruent halves and passes through the vertex
Intercepts
The x-intercepts are the points where the parabola intersects the x-axis, and the y-intercept is where it crosses the y-axis
There are four primary forms of quadratic graphs: standard form, vertex form, factored form, and horizontal parabola
To graph a quadratic function, one must first understand the quadratic equation in its standard form, \(y = ax^2 + bx + c\)
The vertex can be found using the formula \((-b/2a, f(-b/2a))\) or by completing the square to convert the equation into vertex form, and the axis of symmetry is the vertical line \(x = -b/2a\)
After determining the vertex and axis of symmetry, one should calculate the x- and y-intercepts and plot these key points on a coordinate plane
The parabola should be sketched, ensuring it is symmetrical about the axis of symmetry and reflects the correct orientation and width as indicated by the coefficient \(a\)
Quadratic graphs are used in engineering to model forces, trajectories, and electronic circuits
In economics, quadratic functions are used to model cost and revenue functions, enabling businesses to analyze profit maximization and cost minimization
Quadratic graphs are also applied in finance to calculate loan repayments and investment returns, making them valuable tools for financial analysis and decision-making
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Meaning of vertex in quadratic graphs
Vertex is the max or min point of the parabola, indicating the peak or trough.
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Role of coefficient 'a' in parabolas
Coefficient 'a' determines direction of opening (positive for upwards, negative for downwards) and graph's 'sharpness'.
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Intercepts in quadratic graphs
X-intercepts are where graph crosses x-axis; y-intercept is where it crosses y-axis.
