The equation of a curve is y = x^2 + ax + b where a and b are integers. The points (0,-5) and (5,0) lie on the curve. Find the coordinates of the turning point of the curve.

In the equation for our line we have 2 unknowns: a and b. However, we know that the line passes through two known points with x and y coordinates. Therefore, we can begin by substituting in our known x and y coordinates to see if we can find a value for a and b. Substituting in the point (0, -5), we find:-5 = (0)² + (0)a + bTherefore b = -5. Now that we know b, we can substitute in this value for b in to our equation. Using our second set of x and y coordinates, we can find a:0 = (5)² + (5)a - 5which rearranges to give a = -4. Therefore, the equation for our line looks like:y = x² - 4x - 5
From our knowledge of graph sketching, we know that x² (or second order polynomial) graphs are symmetrical about their turning point (as seen in the figure that will be provided during our session). Therefore, if we can find the 2 points at which this line intercepts the x-axis (i.e. when y = 0), we can find the x-coordinate of the halfway point between the two which we can use to find the turning point. We can do this by factorising the equation we have in to the form y=(x+c)(x+d). If one of these brackets equals zero, then y will equal zero, meaning we will have found the x-coordinate of our x-axis intercept. We have already been given one of these points, (5,0), therefore one of these brackets will be (x-5), as when x = 5, this bracket will equal zero and therefore y will equal zero. From inspection, we can see that for the factorised form to equal the original equation, the second bracket is (x+1), giving y = (x+1)(x-5)This can be explained further in the session if required. Therefore, we know that the line intercepts the x-axis at (-1,0) and (5,0). The halfway point between -1 and 5 is 2. Substituting x = 2 in to the equation, y =-9. Therefore, the turning point of this line is at (2,-9).