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What is the range of this quadratic function? $\newline$ $y = x^2 - 2x - 24$ $\newline$ Choices: $\newline$ (A) ${y | y \leq -25}$ $\newline$ (B) ${y | y \geq 1}$ $\newline$ (C) ${y | y \geq -25}$ $\newline$ (D)all real numbers

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Domain and range of quadratic functions: equations

Full solution

Q. What is the range of this quadratic function?

\newline

y = x^2 - 2x - 24

\newline

Choices:

\newline

(A)

{y | y \leq -25}

\newline

(B)

{y | y \geq 1}

\newline

(C)

{y | y \geq -25}

\newline

(D)all real numbers

Find Vertex: We have the quadratic function $y = x^2 - 2x - 24$ . To find the range, we need to determine the vertex of the parabola. The $x$ -coordinate of the vertex can be found using the formula $x = -\frac{b}{2a}$ , where $a$ is the coefficient of $x^2$ and $b$ is the coefficient of $x$ . In this case, $a = 1$ and $b = -2$ .
Calculate x-coordinate: Calculate the x-coordinate of the vertex using the formula $x = -\frac{b}{2a}$ . $\newline$ $x = -\frac{-2}{2\cdot 1}$ $\newline$ $x = \frac{2}{2}$ $\newline$ $x = 1$
Calculate y-coordinate: Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting $x = 1$ into the original equation. $y = (1)^2 - 2(1) - 24$ $y = 1 - 2 - 24$ $y = -25$
Determine Vertex: The vertex of the parabola is $(1, -25)$ . Since the coefficient of $x^2$ is positive $(a = 1)$ , the parabola opens upwards. $\newline$ This means that the vertex represents the minimum point on the graph of the quadratic function.
Find Range: Since the parabola opens upwards and the y-coordinate of the vertex is $-25$ , the range of the function is all y-values greater than or equal to $-25$ . $\newline$ Range: $\{y \mid y \geq -25\}$

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