Textbook Question
Define the quadratic function ƒ having x-intercepts (1, 0) and (-2, 0) and y-intercept(0, 4).
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4. Polynomial Functions
Quadratic Functions
2:35 minutesProblem 98
Textbook Question
In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-1) The graph passes through the point (-2,-3).
Verified step by step guidance1
insert step 1: Start with the vertex form of a parabola's equation, which is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
insert step 2: Substitute the vertex (-3, -1) into the equation, replacing h with -3 and k with -1, to get y = a(x + 3)^2 - 1.
insert step 3: Use the point (-2, -3) that the parabola passes through to find the value of a. Substitute x = -2 and y = -3 into the equation: -3 = a(-2 + 3)^2 - 1.
insert step 4: Simplify the equation to solve for a. This involves calculating (-2 + 3)^2, which is 1, and then solving the equation -3 = a(1) - 1.
insert step 5: Solve for a by adding 1 to both sides to get -2 = a, which gives you the value of a. Now, substitute a back into the vertex form equation to get the final equation of the parabola.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertex Form of a Parabola
The vertex form of a parabola is expressed as y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form is particularly useful for graphing and understanding the transformations of the parabola, as it clearly indicates the vertex's position and the direction and width of the parabola based on the value of 'a'.
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Vertex Form
Finding 'a' in Vertex Form
To determine the value of 'a' in the vertex form equation, you can substitute a known point on the parabola into the equation. In this case, using the point (-2, -3) along with the vertex (-3, -1) allows you to solve for 'a', which indicates how 'steep' or 'wide' the parabola is.
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Substituting Points into Equations
Substituting points into equations is a fundamental algebraic technique used to find unknown values. By plugging in the x and y coordinates of a point into the equation, you can create an equation that can be solved for the unknown variable, which is essential for determining parameters like 'a' in the vertex form of a parabola.
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