Question: For the quadratic function \( f(x) = x^2 + 5x - 7 \), find the vertex, the axis of symmetry, and…

For the quadratic function \( f(x) = x^2 + 5x - 7 \), find the vertex, the axis of symmetry, and the \( x \)-intercept(s). Express the vertex as a coordinate pair. If there is more than one intercept, enter as a comma-separated list of points.

• vertex:
• axis of symmetry:
• \( x \)-intercept(s):

Solution

For the quadratic function \( f(x) = x^2 + 5x - 7 \), we need to find the vertex, the axis of symmetry, and the x-intercept(s). First, find the vertex. The vertex form of a quadratic function is given by: \[ f(x) = a(x-h)^2 + k \] where \((h, k)\) is the vertex of the parabola. In the standard form \( ax^2 + bx + c \), the x-coordinate of the vertex \( h \) can be found using: \[ h = -\frac{b}{2a} \] Here, \( a = 1 \), \( b = 5 \), and \( c = -7 \). \[ h = -\frac{5}{2 \times 1} = -\frac{5}{2} \] Now, substitute \( h \) into the function to find \( k \). \[ f\left(-\frac{5}{2}\right) = \left(-\frac{5}{2}\right)^2 + 5\left(-\frac{5}{2}\right) - 7 \] \[ = \frac{25}{4} - \frac{25}{2} - 7 \] \[ = \frac{25}{4} - \frac{50}{4} - \frac{28}{4} \] \[ = \frac{25 - 50 - 28}{4} \] \[ = -\frac{53}{4} \] So, the vertex is \( \left(-\frac{5}{2}, -\frac{53}{4}\right) \). Next, find the axis of symmetry, which is the vertical line through the vertex: \[ x = -\frac{5}{2} \] Finally, find the x-intercept(s) by setting \( f(x) = 0 \). \[ x^2 + 5x - 7 = 0 \] Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute \( a = 1 \), \( b = 5 \), \( c = -7 \): \[ x = \frac{-5 \pm \sqrt{5^2 - 4 \times 1 \times (-7)}}{2 \times 1} \] \[ x = \frac{-5 \pm \sqrt{25 + 28}}{2} \] \[ x = \frac{-5 \pm \sqrt{53}}{2} \] The x-intercepts are \( \left( \frac{-5 + \sqrt{53}}{2}, 0 \right) \) and \( \left( \frac{-5 - \sqrt{53}}{2}, 0 \right) \). In summary: - Vertex: \( \left(-\frac{5}{2}, -\frac{53}{4}\right) \) - Axis of symmetry: \( x = -\frac{5}{2} \) - X-intercepts: \( \left( \frac{-5 + \sqrt{53}}{2}, 0 \right), \left( \frac{-5 - \sqrt{53}}{2}, 0 \right) \)