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$Which function represents a quadratic function whose graph has a range of {y∣y >= 3} ? (A) f(x)=x^(2)-6x+3 (B) f(x)=x^(2)-3 (C) f(x)=x^(2)+3 (D) f(x)=x^(2)+6x+3$

$24$ . Which function represents a quadratic function whose graph has a range of $\{y \mid y \geq 3\}$ ? $\newline$ (A) $f(x)=x^{2}-6 x+3$ $\newline$ (B) $f(x)=x^{2}-3$ $\newline$ (C) $f(x)=x^{2}+3$ $\newline$ (D) $f(x)=x^{2}+6 x+3$

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Math Problems

Algebra 1

Domain and range of quadratic functions: equations

Full solution

24

. Which function represents a quadratic function whose graph has a range of

\{y \mid y \geq 3\}

\newline

(A)

f(x)=x^{2}-6 x+3

\newline

(B)

f(x)=x^{2}-3

\newline

(C)

f(x)=x^{2}+3

\newline

(D)

f(x)=x^{2}+6 x+3

Identify Vertex Form: Identify the vertex form of a quadratic function, which is $y = a(x-h)^2 + k$ , where $(h, k)$ is the vertex.
Analyze Option A: Analyze option (A): $f(x) = x^2 - 6x + 3$ . Complete the square to find the vertex form. $\newline$ $f(x) = (x^2 - 6x + 9) - 9 + 3 = (x - 3)^2 - 6$ . Vertex is $(3, -6)$ .
Analyze Option B: Since the vertex $(3, -6)$ indicates the minimum point and the parabola opens upwards $(a > 0)$ , the range is ${y|y \geq -6}$ . This does not match ${y|y \geq 3}$ .
Analyze Option C: Analyze option (B): $f(x) = x^2 - 3$ . This is already in vertex form with vertex $(0, -3)$ .
Analyze Option D: The vertex $(0, -3)$ shows the minimum point, and the parabola opens upwards. The range is ${y|y \geq -3}$ . This does not match ${y|y \geq 3}$ .
Analyze Option D: The vertex $(0, -3)$ shows the minimum point, and the parabola opens upwards. The range is ${y|y \geq -3}$ . This does not match ${y|y \geq 3}$ . Analyze option (C): $f(x) = x^2 + 3$ . This is in vertex form with vertex $(0, 3)$ .
Analyze Option D: The vertex $(0, -3)$ shows the minimum point, and the parabola opens upwards. The range is $\{y|y \geq -3\}$ . This does not match $\{y|y \geq 3\}$ . Analyze option (C): $f(x) = x^2 + 3$ . This is in vertex form with vertex $(0, 3)$ . The vertex $(0, 3)$ indicates the minimum point, and the parabola opens upwards. The range is $\{y|y \geq 3\}$ . This matches the required range.
Analyze Option D: The vertex $(0, -3)$ shows the minimum point, and the parabola opens upwards. The range is ${y|y \geq -3}$ . This does not match ${y|y \geq 3}$ . Analyze option (C): $f(x) = x^2 + 3$ . This is in vertex form with vertex $(0, 3)$ . The vertex $(0, 3)$ indicates the minimum point, and the parabola opens upwards. The range is ${y|y \geq 3}$ . This matches the required range. Analyze option (D): $f(x) = x^2 + 6x + 3$ . Complete the square to find the vertex form. $\newline$ $f(x) = (x^2 + 6x + 9) - 9 + 3 = (x + 3)^2 - 6$ . Vertex is $(-3, -6)$ .
Analyze Option D: The vertex $(0, -3)$ shows the minimum point, and the parabola opens upwards. The range is ${y|y \geq -3}$ . This does not match ${y|y \geq 3}$ . Analyze option (C): $f(x) = x^2 + 3$ . This is in vertex form with vertex $(0, 3)$ . The vertex $(0, 3)$ indicates the minimum point, and the parabola opens upwards. The range is ${y|y \geq 3}$ . This matches the required range. Analyze option (D): $f(x) = x^2 + 6x + 3$ . Complete the square to find the vertex form. $\newline$ $f(x) = (x^2 + 6x + 9) - 9 + 3 = (x + 3)^2 - 6$ . Vertex is $(-3, -6)$ . Since the vertex $(-3, -6)$ indicates the minimum point and the parabola opens upwards, the range is ${y|y \geq -3}$ $1$ . This does not match ${y|y \geq 3}$ .