Maths/Gr11 Investigation NW/Feb 2025 D1.4 How many times will the line \( y=-2 x+\frac{9}{2} \) intersect with graph of \( g \). D1.5 Determine the values of \( k \) for which line \( y=-2 x+k \) will not intersect with \( g \). \( \qquad \) [7]

To determine how many times the line \( y = -2x + \frac{9}{2} \) intersects with the graph of \( g \), you should set \( g = -2x + \frac{9}{2} \) and solve for \( x \). This will show how many solutions (or intersection points) exist. If there are two solutions, the line intersects at two points; if one, then it's a tangent; and if none, they don’t intersect at all! For the line \( y = -2x + k \) to not intersect with \( g \), the slopes of both lines must be equal, but the y-intercepts must be different. This means you need to determine the critical point where \( k \) is either too high or too low, putting the lines completely apart. By analyzing the conditions where the discriminant of the resulting quadratic (from substituting \( g \) into the line equation) is less than zero, you can pinpoint the values of \( k \)!