What can we determine about the function h(x) = x² + 20x + 17? Please check all applicable statements.

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The function h(x) = x² + 20x + 17 has several important characteristics. Here are some key points to consider:

  • It is a quadratic function: This function is a polynomial of degree 2, which means its graph will be a parabola. Quadratic functions have the general form of ax² + bx + c, where a, b, and c are constants.
  • The parabola opens upwards: Since the coefficient of x² (which is 1 in this case) is positive, the parabola will open upwards.
  • The vertex: The vertex form of a quadratic function can be found by calculating the x-coordinate of the vertex using the formula x = -b / (2a). For this function, a = 1 and b = 20, leading to x = -20 / (2 * 1) = -10. Plugging this back into the function gives the y-coordinate of the vertex as h(-10) = (-10)² + 20(-10) + 17 = 100 - 200 + 17 = -83. Therefore, the vertex is at (-10, -83).
  • The y-intercept: The y-intercept is the point where the graph intersects the y-axis. We can find this by evaluating h(0), which gives h(0) = 0² + 20(0) + 17 = 17. Thus, the y-intercept is at (0, 17).
  • The axis of symmetry: The axis of symmetry of a parabola can be found at x = -b / (2a). Therefore, the axis of symmetry for this function is x = -10.
  • Minimum value: Since the parabola opens upwards, the vertex at (-10, -83) represents the minimum point of the function. Hence, the minimum value of h(x) occurs at h(-10) = -83.

In summary, the function h(x) = x² + 20x + 17 is a quadratic function that opens upwards, has a vertex at (-10, -83), a y-intercept at (0, 17), and has an axis of symmetry at x = -10. All these attributes provide important insights into the shape and behavior of the function.

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