ACTIVITY \( 6: ~\{2 \) Marks \( \} \) Explain why the roots of \( a x^{2}+b x+c=0 \) will be the \( x \)-intercepts of the parabola.

To understand why the roots of the quadratic equation \( ax^2 + bx + c = 0 \) correspond to the \( x \)-intercepts of the parabola represented by the equation, we can follow these steps: 1. **Definition of Roots**: The roots of the equation \( ax^2 + bx + c = 0 \) are the values of \( x \) for which the equation equals zero. In other words, if \( r_1 \) and \( r_2 \) are the roots, then: \[ a r_1^2 + b r_1 + c = 0 \] \[ a r_2^2 + b r_2 + c = 0 \] 2. **Graph of the Quadratic Function**: The equation \( y = ax^2 + bx + c \) represents a parabola in the Cartesian coordinate system. The shape of the parabola (opening upwards or downwards) depends on the coefficient \( a \): - If \( a > 0 \), the parabola opens upwards. - If \( a < 0 \), the parabola opens downwards. 3. **Finding \( x \)-Intercepts**: The \( x \)-intercepts of a graph are the points where the graph crosses the \( x \)-axis. At these points, the value of \( y \) is zero. Therefore, to find the \( x \)-intercepts, we set the equation of the parabola equal to zero: \[ ax^2 + bx + c = 0 \] 4. **Conclusion**: Since the roots \( r_1 \) and \( r_2 \) are the solutions to the equation \( ax^2 + bx + c = 0 \), they represent the \( x \)-values where the parabola intersects the \( x \)-axis. Thus, the roots of the quadratic equation are indeed the \( x \)-intercepts of the parabola. In summary, the roots of the quadratic equation \( ax^2 + bx + c = 0 \) correspond to the \( x \)-intercepts of the parabola because they are the values of \( x \) that make the equation equal to zero, indicating where the graph crosses the \( x \)-axis.