Chapter 3 Shapes of Quadratic Graphs In this section, our main aim is to (i) find out the signs of a, b and c from the given graph of y=ax 2 +bx+c, (ii) sketch the graph of y=ax 2 +bx+c if the sign of a, b and c are given

Part I Sign of a We have the following two situations : Shapes of Quadratic Graphs

When a > 0 The curve is opening upwards. y=ax 2 +bx+c Shapes of Quadratic Graphs

When a < 0 The curve is opening downwards. y=ax 2 +bx+c Shapes of Quadratic Graphs

Definition Given the graph of y = ax 2 +bx+c y=ax 2 +bx+c Part II x and y-intercepts, the value of c the value of the x-coordinate(s) of the point(s) where the curve cuts the x-axis x-intercept(s) = Shapes of Quadratic Graphs

Definition y=ax 2 +bx+c Shapes of Quadratic Graphs y-intercept =the value of the y-coordinate of the point where the curve cuts the y-axis

Facts Given the graph of y = ax 2 +bx+c. We can observe that : y=ax 2 +bx+c 1. To find x-intercepts, we always put y = 0 into the equation of the curve and solve for x. Shapes of Quadratic Graphs

Facts y=ax 2 +bx+c 2. To find y-intercept, we always put x = 0 into the equation of the curve and solve for y. Shapes of Quadratic Graphs

Fact 1 Given the graphs of y = ax 2 +bx+c. We can observe that : The curve has ONE y-intercept By putting x = 0, we find that y-intercept = c Shapes of Quadratic Graphs

Fact 2 The curve has TWO x-interceptsifb 2 - 4ac > 0 ONE x-interceptifb 2 - 4ac = 0 NO x-interceptifb 2 - 4ac < 0 Shapes of Quadratic Graphs

No x-intercept b 2 - 4ac < 0 Two x-intercepts b 2 - 4ac > 0 One x-intercept b 2 - 4ac = 0

By putting y = 0, we find that x-intercept(s) = the root(s) of the quadratic equation ax 2 +bx+c=0 Shapes of Quadratic Graphs Fact 3 x-intercept(s) = the roots of the quadratic equation ax 2 +bx+c=0

Given the graph of y = ax 2 +bx+c Part III Sign of b Hence we have From the above, we know that x-intercepts = roots of the equation ax 2 +bx+c =0 sum of x-intercepts = - b / a Shapes of Quadratic Graphs Sum of these two values = - b / a Once we know the value of a, then we can use it to find the value of b.

Line of symmetry is x = - b / 2a Alternate Method : Recall : Given the graph of y = ax 2 +bx+c, we have Line of symmetry x = - b / 2a Shapes of Quadratic Graphs

Again if we know the value of a, then we can use it to find the value of b. So we have the following facts : - b / 2a is positive if th e line of symmetry lies on the right of y-axis - b / 2a is negative if th e line of symmetry lies on the left of y-axis Shapes of Quadratic Graphs

Let us take a look at the following simple example. (1) It is opening upwards. So a > 0. (2) Since y-intercept is negative, c < 0. y=ax 2 +bx+c Shapes of Quadratic Graphs Given the graph of y=ax 2 +bx+c

Shapes of Quadratic Graphs (3) From the graph, sum of x-intercepts is negative. y=ax 2 +bx+c Therefore – b/a is negative. So – b is negative. (by (1), a is positive) Then b is positive.

Shapes of Quadratic Graphs (3) Alternative Method : The line of symmetry lies on the left of y-axis. Therefore – b/2a is negative. Similar to the above, we can conclude that b is positive. Line of symmetry

Shapes of Quadratic Graphs Conclusion : a > 0 b > 0 c < 0 y=ax 2 +bx+c

We can use some different types of software to show the graphs of the quadratic curves. For different values of a, b and c, we have different shapes of y=ax 2 +bx+c. Part IV Using computer software Shapes of Quadratic Graphs

1. Microsoft Excel 3. Graphmatica2. Sketchpad Shapes of Quadratic Graphs Here are some examples of software :

Example (a) Sketch the graph of y = ax 2 + c, where a 0. Part V Examples (b) Find the signs of a, b and c : Shapes of Quadratic Graphs

Solution (a) Step 1: since a < 0, the curve is opening downwards Step 2 : y-intercept = c = positive Step 3 : Line of symmetry is x = -b / 2a = 0. So the line of symmetry is the y-axis. Shapes of Quadratic Graphs

From the above analysis, we can draw the graph : Shapes of Quadratic Graphs

(b) Step 1 : Since the curve is opening downwards, a < 0. Step 2 : Since y-intercept is positive, c > 0. Step 3 : The sum of x-intercepts is negative, so -b / a is negative. Since a is negative, b must be negative. Conclusion : a 0 Shapes of Quadratic Graphs