The expression x^{2} + bx + c is called a quadratic function of x, where a, b and c are constants. When a quadratic function y = x2 + bx + c is plotted, its graph gives a smooth curve called a parabola.

Consider \( \scriptsize y = x^2 + bx + c\)

Also, we can have Consider \( \scriptsize y = -ax^2 + bx + c\)

**Hint: **The highest power of x is 2 i.e. a second degree function

- The points T
_{1}and T_{2}are the turning points i.e. where the curve changes direction (inflexion point) - When a is positive, we have a minimum curve with shape and the minimum value of the function y
_{min}is at the turning point T_{1}and the value of x where it occurs gives the equation of the line of symmetry. - When a is negative, we have a maximum curve shape and the maximum value of the function y
_{max}is at the turning point T_{2}and the value of x where it occurs gives the equation of the line of symmetry. - The line of symmetry or the axis of symmetry divides the curve into two equal parts.

Example 1

Plot the curve of y = x^{2} + x â€“ 6 for values of x from -4 to 3

(a) Use the curve to find :

(i) the values of x when y = 2.8

(ii) the value of y when x = 1.4

(iii) the minimum value of the function and the value of x

(b) Draw the axis of symmetry of the curve and write down its equation

**Solution:**

X | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |

X^{2} | 16 | 9 | 4 | 1 | 0 | 1 | 4 | 9 |

X | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |

-6 | -6 | -6 | -6 | -6 | -6 | -6 | -6 | -6 |

Y | 6 | 0 | -4 | -6 | -6 | -4 | 0 | 6 |

(a) From the graph:

(i) when y = 2.8, x= -3.5 or x = -2.6

(ii) when x = 1.4, y = -2.6

(iii) y_{min.} = -6.4 at x = -0.5

(b) Equation axis of symmetry: x = \(– \frac {1}{2} \)

**Scale:** On x-axis 1unit â‰¡1cm

Â Â Â Â Â Â Â Â Â Â Â On y-axis 1unit â‰¡ 1cm

Example 2

Draw the curve of y = -x^{2} for values of x from -4 to 4.

Use the curve to find:

(i) The maximum value of the curve

(ii) the equation of the line of symmetry

(iii) the value of x when y=-10

(iv) The value of y when x = 2.5

X | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |

Y | -16 | -9 | -4 | -1 | 0 | -1 | -4 | -9 | -16 |

From the curve:

(i) Maximum value =0 i.e. at the origin we have the turning point.

(ii) The curve is symmetric about the y-axis thus the equation of the line of symmetry is x = 0

(iii) When y = -10, x = 3.2 or -3.2

(iv) When x = 2.5, y = 6.2

Scale: 1unit =1cm on x-axis

2units = 1cm on y-axis

Example 3

(a) Copy and complete the table of values ofÂ y = 3 – 4x – 2x^{2}

X | -4 | -3 | -2 | -1 | 0 | 1 | 2 |

Y |

(b) Using a scale of 1cm to 1unit on the x-axis and 1cm to 2units on the y-axis, plot the graph of y = 3 – 4x – 2x^{2} for values of x from -4 to 2

(c) Use the graph to find :

(i) the highest value of the function 3 – 4x – 2x^{2}

(ii) the two values of x when y=-10

(iii)The coordinates of the point where y is

(d) Draw the line of symmetry and state its equation

(e) For what range of values of x is y > 0?

**Solution: (a)**

X | -4 | -3 | -2 | -1 | 0 | 1 | 2 |

Y | -13 | -3 | 3 | 5 | 3 | -3 | -13 |

(b)

(c) (i) y_{max} = 5

(ii) x = -3.7 or x = 1.7

(iii) Coordinate of y_{max} = (-1.5)

(d) Equation of line of symmetry: x = -1

(e) Range of x for y > 0; -2.5< x < 0.5

Scale: 1unit â‰¡ 1cm on x-axis | 2units â‰¡ 1cm on y-axis

Example 4

(a) Draw the graph of y = 2x^{2} -3x – 5 for -3 â‰¤ x â‰¤ 4 and hence use the graph to solve the equation 2x^{2} – 3x – 5 = 0

(b) Use your graph to solve the equation 2x^{2} + x -2 = 4x + 8

**Solution:**

X | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |

2x^{2} | 18 | 8 | 2 | 0 | 2 | 8 | 18 | 32 |

-3x | 9 | 6 | 3 | 0 | -3 | -6 | -9 | -12 |

-5 | -5 | -5 | -5 | -5 | -5 | -5 | -5 | -5 |

y | 22 | 9 | 0 | -5 | -6 | -3 | 4 | 15 |

Scale: 1unit â‰¡ 1cm on x-axis. Â Â 5units â‰¡ 1cm on y-axis

(a) The roots of the equation at point where the graph cuts the x-axis i.e. x = -1 or x = 2.5

(b) 2x^{2} + x -2 = 4x + 8

Subtract 4x from both sides

i.e. 2x^{2} – 3x – 2 = 8

subtract 3 from both sides

i.e. 2x^{2} – 3x – 5 = 5

i.e. y = 5

From the graph x = -1.6 or x = 3.2

Scale: 1unit â‰¡ 1cm on x-axis and 5units â‰¡ 1cm on y-axis

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