91Ó°ÊÓ

The vertex of a parabola is an essential point that represents either the highest or lowest point on the graph of a quadratic function, depending on whether the parabola opens upwards or downwards. For a quadratic function in the form \( f(x) = ax^2 + bx + c \), the formula to find the x-coordinate of the vertex is given by \( h = \frac{-b}{2a} \). By plugging in the values from our exercise, \( a = 1 \) and \( b = 1 \), we find \( h = \frac{-1}{2 \cdot 1} = -\frac{1}{2} \). The y-coordinate is found by substituting \( h \) into the original function: \( k = f(h) \). For \( f(x)=x^{2}+x-1 \), this gives us \( k = f(-\frac{1}{2}) = \left(-\frac{1}{2}\right)^2 - \frac{1}{2} - 1 = -\frac{5}{4} \). Thus, the vertex is at \( \left(-\frac{1}{2}, -\frac{5}{4}\right) \), which is a point of minimum because the parabola opens upwards.

The y-intercept of a parabola is the point where the graph crosses the y-axis. To find this point, simply set \( x = 0 \) in the quadratic function and solve for \( f(x) \). This gives you the y-coordinate of the intercept. For our function \( f(x) = x^{2} + x - 1 \), setting \( x = 0 \) yields \( f(0) = 0^{2} + 0 - 1 = -1 \). Therefore, the y-intercept is at \( (0, -1) \). This tells us where our parabola hits the y-axis, providing a fixed point in our graph.

The x-intercepts are where the graph of the quadratic function crosses the x-axis. These intercepts occur when \( f(x) = 0 \). To find the x-intercepts, we solve the equation \( x^{2} + x - 1 = 0 \). Since factoring directly might be complicated, we use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).Plugging in \( a = 1 \), \( b = 1 \), and \( c = -1 \) into the formula yields:\( x = \frac{-1 \pm \sqrt{(1)^2 - 4(1)(-1)}}{2(1)} = \frac{-1 \pm \sqrt{5}}{2} \).These calculations show the x-intercepts at \( \left(\frac{-1+\sqrt{5}}{2}, 0\right) \) and \( \left(\frac{-1-\sqrt{5}}{2}, 0\right) \). The x-intercepts are crucial for understanding where the graph intersects the x-axis.

The quadratic formula is a powerful tool for finding the roots or x-intercepts of quadratic equations, especially when factoring is not practical. The formula is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).This formula provides both the x-intercepts of the equation when the discriminant \( b^2 - 4ac \) is greater than or equal to zero.

• If the discriminant is positive, there are two distinct real x-intercepts.
• If it is zero, there is exactly one real x-intercept (the vertex lies on the x-axis).
• If negative, there are no real x-intercepts, and the parabola does not cross the x-axis.

In our exercise, with \( a = 1 \), \( b = 1 \), \( c = -1 \), the discriminant \( (1)^2 - 4(1)(-1) = 5 \) is positive, indicating two distinct real roots.Learning to use the quadratic formula gives a consistent method to determine the x-intercepts of any quadratic equation.