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The x-intercepts of a quadratic function are the points on the graph where the curve crosses the x-axis. At these points, the value of the function, also known as the y-value or f(x), is zero. To find the x-intercepts, we set the quadratic equation equal to zero and solve for x. These solutions represent where the parabola touches or cuts through the x-axis.

In the problem, the quadratic function is given by \( f(x) = -x^2 - 10x - 21 \). To find the x-intercepts, you need to determine the values of x for which \( f(x) = 0 \). This involves solving the equation \( -x^2 - 10x - 21 = 0 \). By solving this equation, you would find the points where the graph intersects the x-axis, yielding the x-intercepts.

The y-intercept of a function is the point where the graph crosses the y-axis. At this point, the value of x is zero. To find the y-intercept of a quadratic function, we substitute 0 for x in the function, and then solve for \( f(x) \).

For the quadratic function \( f(x) = -x^2 - 10x - 21 \), substitute 0 for x:

• \( -f(0) = -0^2 - 10 \times 0 - 21 = -21 \)

Thus, the graph intersects the y-axis at the point (0, -21), which represents the y-intercept of the function. This point tells us where the parabola touches the y-axis.

The quadratic formula is a reliable method for finding the x-intercepts of any quadratic function. A quadratic equation is generally in the form \( ax^2 + bx + c = 0 \). The quadratic formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In this formula, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation.

Using the quadratic function \( f(x) = -x^2 - 10x - 21 \), we identify:

• \( a = -1 \)
• \( b = -10 \)
• \( c = -21 \)

These values are then substituted into the quadratic formula to find the x-intercepts, helping us determine where the quadratic graph crosses the x-axis.

The discriminant is a component of the quadratic formula that can tell us about the nature of the roots of a quadratic equation. It is the part under the square root: \( b^2 - 4ac \). Depending on its value, the discriminant indicates the number and type of solutions:

• If the discriminant is positive, there are two distinct real solutions (two x-intercepts).
• If the discriminant is zero, there is exactly one real solution (the vertex of the parabola touches the x-axis).
• If the discriminant is negative, there are no real solutions (the parabola does not cross the x-axis).

For \( f(x) = -x^2 - 10x - 21 \), the discriminant is calculated as:

• \((-10)^2 - 4 \times (-1) \times (-21) = 100 - 84 = 16 \)

Since the discriminant is 16, which is positive, it confirms that the quadratic function has two real x-intercepts. This gives us confidence that solving the quadratic equation will yield two distinct x-values where the graph crosses the x-axis.