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Chapter 8: Problem 35

Find a quadratic function that fits the set of data points. $$(1,4),(-1,-2),(2,13)$$

Short Answer

Expert verified
The quadratic function is \( f(x) = 2x^2 + 3x - 1 \).

Step by step solution

01

- Write the General Form of a Quadratic Function

A quadratic function can be written in the form: \[ f(x) = ax^2 + bx + c \] where \(a\), \(b\), and \(c\) are constants we need to find.
02

- Set Up the System of Equations

Substitute each given point into the quadratic function to form a system of equations.For the point \((1, 4)\): \[ a(1)^2 + b(1) + c = 4 \] which simplifies to: \[ a + b + c = 4 \]For the point \((-1, -2)\): \[ a(-1)^2 + b(-1) + c = -2 \] which simplifies to: \[ a - b + c = -2 \]For the point \((2, 13)\): \[ a(2)^2 + b(2) + c = 13 \] which simplifies to: \[ 4a + 2b + c = 13 \]
03

- Solve the System of Equations

We now have the following system of linear equations: \[1) \, a + b + c = 4 \2) \, a - b + c = -2 \3) \, 4a + 2b + c = 13 \]Subtract equation 2 from equation 1: \[ (a + b + c) - (a - b + c) = 4 - (-2) \]\[ 2b = 6 \]\[ b = 3 \] Now, subtract equation 1 from equation 3: \[ (4a + 2b + c) - (a + b + c) = 13 - 4 \]\[ 3a + b = 9 \] Substitute \(b = 3\) into \(3a + b = 9\): \[ 3a + 3 = 9 \]\[ 3a = 6 \]\[ a = 2 \] Finally, substitute \(a = 2\) and \(b = 3\) into equation 1: \[ 2 + 3 + c = 4 \]\[ 5 + c = 4 \]\[ c = -1 \]
04

- Write the Quadratic Function

Using the values of \(a\), \(b\), and \(c\) we found, we can write the quadratic function as: \[ f(x) = 2x^2 + 3x - 1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving quadratic equations
Solving quadratic equations is a fundamental concept in algebra. A quadratic equation can be written in the form ax2 + bx + c = 0, where a, b and c are constants.

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Most popular questions from this chapter

Explain how the leading coefficient of a quadratic function can be used to determine whether a maximum or a minimum function value exists.

Solve each formula for the indicated letter. Assume that all variables represent positive numbers. \(a^{2}+b^{2}=c^{2},\) for \(b\) (Pythagorean formula in two dimensions)

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (-4,-2) $$

Solve each formula for the indicated letter. Assume that all variables represent positive numbers. \(a^{2}+b^{2}+c^{2}=d^{2},\) for \(c\) (Pythagorean formula in three dimensions)

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (2,8) $$
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