/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 88 Suppose \(f(x)=a x^{2}+b x+c,\) ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 88

Suppose \(f(x)=a x^{2}+b x+c,\) where \(a \neq 0 .\) Show that the vertex of the graph of \(f\) is the point \(\left(-\frac{b}{2 a}, \frac{4 a c-b^{2}}{4 a}\right)\).

Short Answer

Expert verified
The vertex of the quadratic function \(f(x) = ax^2 + bx + c\) with \(a \neq 0\) is given by the point \(\left(-\frac{b}{2a}, \frac{4ac-b^2}{4a}\right)\).

Step by step solution

01

Find the x-coordinate of the vertex

The general form of a quadratic function is \(f(x) = ax^2 + bx + c\). To find the x-coordinate of the vertex, use the formula \(\frac{-b}{2a}\): \( x_v = -\frac{b}{2a} \)
02

Find the y-coordinate of the vertex

Now, to find the y-coordinate of the vertex, substitute the x-coordinate found in Step 1 into the given function: \( y_v = f\left(-\frac{b}{2a}\right) = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c \)
03

Simplify the equation for the y-coordinate

Simplify the expression to determine the y-coordinate of the vertex: \( y_v = a\left(\frac{b^2}{4a^2}\right) - b\left(\frac{b}{2a}\right) + c \) Multiplying the terms inside the brackets and combining like terms, we get: \( y_v = \frac{ab^2}{4a^2} - \frac{b^2}{2a} + c \) Now, find a common denominator for the fractions: \( y_v = \frac{2ab^2 - 4ab^2 + 8a^2c}{4a^2} \) Combine the terms in the numerator: \( y_v = \frac{-2ab^2 + 8a^2c}{4a^2} \) Finally, simplify the fraction: \( y_v = \frac{4ac-b^2}{4a} \)
04

Find the vertex

Now that we have both x-coordinate and y-coordinate of the vertex, we can express the vertex as a point (x, y): Vertex = \(\left(-\frac{b}{2a}, \frac{4ac-b^2}{4a}\right)\)

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