Questions: Identifying The Domain, Range and Horizontal Intercepts of a Quadratic Function For each of the following quadratic functions: 1. Determine the Domain of the Function 2. Determine the Range of the Function 3. Use the Intersect Feature and your graphing calculator to determine the Horizontal Intercepts, if any. Round your answers to one decimal places as needed [Hint: If two Horizontal Intercepts exist, enter them as (x1, y1),(x2, y2). If only one exists, enter (x1, y1). If none exist, enter DNE] Function Domain In Interval Notation Range In Interval Notation Horizontal Intercepts As a list of ordered pairs f(x)=2 x^2+4 x-9 (-infty, infty) [-11, infty) (1.3,0),(-3.3, 0) g(x)=x^2+8 x+21 (-infty, infty) [5, infty) DNE (x)=-x^2+8 (-infty, infty) (-infty, 8] (-2.8,0),(2.8, 0) p(t)=3 t^2-12 t (-infty, infty) [-12, infty) h(x)=2 x^2 (-infty, infty) [0, infty)

Identifying The Domain, Range and Horizontal Intercepts of a Quadratic Function For each of the following quadratic functions: 1. Determine the Domain of the Function 2. Determine the Range of the Function 3. Use the Intersect Feature and your graphing calculator to determine the Horizontal Intercepts, if any. Round your answers to one decimal places as needed [Hint: If two Horizontal Intercepts exist, enter them as (x1, y1),(x2, y2). If only one exists, enter (x1, y1). If none exist, enter DNE] Function Domain In Interval Notation Range In Interval Notation Horizontal Intercepts As a list of ordered pairs f(x)=2 x^2+4 x-9 (-infty, infty) [-11, infty) (1.3,0),(-3.3, 0) g(x)=x^2+8 x+21 (-infty, infty) [5, infty) DNE (x)=-x^2+8 (-infty, infty) (-infty, 8] (-2.8,0),(2.8, 0) p(t)=3 t^2-12 t (-infty, infty) [-12, infty) h(x)=2 x^2 (-infty, infty) [0, infty)
Transcript text: Identifying The Domain, Range and Horizontal Intercepts of a Quadratic Function For each of the following quadratic functions: 1. Determine the Domain of the Function 2. Determine the Range of the Function 3. Use the Intersect Feature and your graphing calculator to determine the Horizontal Intercepts, if any. Round your answers to one decimal places as needed [Hint: If two Horizontal Intercepts exist, enter them as $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$. If only one exists, enter $\left(x_{1}, y_{1}\right)$. If none exist, enter DNE] \begin{tabular}{|c|c|c|c|} \hline Function & \begin{tabular}{l} Domain \\ In Interval Notation \end{tabular} & \begin{tabular}{l} Range \\ In Interval Notation \end{tabular} & Horizontal Intercepts As a list of ordered pairs \\ \hline $f(x)=2 x^{2}+4 x-9$ & $(-\infty, \infty)$ & $[-11, \infty)$ & $(1.3,0),\left(-3.3, 0\right)$ \\ \hline $g(x)=x^{2}+8 x+21$ & $(-\infty, \infty)$ & $[5, \infty)$ & DNE \\ \hline $(x)=-x^{2}+8$ & $(-\infty, \infty)$ & $(-\infty, 8]$ & $(-2.8,0),\left(2.8, 0\right)$ \\ \hline $p(t)=3 t^{2}-12 t$ & $(-\infty, \infty)$ & $[-12, \infty)$ & \\ \hline $h(x)=2 x^{2}$ & $(-\infty, \infty)$ & $[0, \infty)$ & \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Find the Domain of the first three functions

The domain of a quadratic function is all real numbers. In interval notation, this is represented as (-∞, ∞).

Step 2: Find the Range of the first three functions

  • f(x) = 2x² + 4x - 9: The vertex occurs at x = -b/2a = -4/(2*2) = -1. f(-1) = 2(-1)² + 4(-1) - 9 = -11. Since the parabola opens upwards (a > 0), the range is [-11, ∞).

  • g(x) = x² + 8x + 21: The vertex occurs at x = -b/2a = -8/(2*1) = -4. g(-4) = (-4)² + 8(-4) + 21 = 5. Since the parabola opens upwards (a > 0), the range is [5, ∞).

  • f(x) = -x² + 8: The vertex occurs at x = -b/2a = -0/(2*-1) = 0. f(0) = -(0)² + 8 = 8. Since the parabola opens downwards (a < 0), the range is (-∞, 8].

Step 3: Find the Horizontal Intercepts of the first three functions

  • f(x) = 2x² + 4x - 9: Set f(x) = 0 and solve for x using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a = (-4 ± √(16 - 4(2)(-9))) / (2*2) = (-4 ± √88) / 4. This gives approximately x = 1.3 and x = -3.3. So, the intercepts are (1.3, 0) and (-3.3, 0).

  • g(x) = x² + 8x + 21: Set g(x) = 0 and solve for x. The discriminant is b² - 4ac = 8² - 4(1)(21) = 64 - 84 = -20. Since the discriminant is negative, there are no real roots, and thus no horizontal intercepts. The answer is DNE.

  • f(x) = -x² + 8: Set f(x) = 0 and solve for x: -x² + 8 = 0 => x² = 8 => x = ±√8 ≈ ±2.8. The intercepts are (-2.8, 0) and (2.8, 0).

Final Answer:

  1. f(x) = 2x² + 4x - 9: Domain: (-∞, ∞), Range: [-11, ∞), Horizontal Intercepts: (1.3, 0), (-3.3, 0)
  2. g(x) = x² + 8x + 21: Domain: (-∞, ∞), Range: [5, ∞), Horizontal Intercepts: DNE
  3. f(x) = -x² + 8: Domain: (-∞, ∞), Range: (-∞, 8], Horizontal Intercepts: (-2.8, 0), (2.8, 0)
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