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Curve Sketching - Introduction

In this introduction, we will introduce the key features of a curve to help sketching and look at the simple case of quadratic curves in relation to these properties.

Features of a curve

When sketching a curve, the following properties must be considered

  1. y-intercept
  2. gradient near origin
  3. roots
  4. turning points
  5. asymptotes
  6. behaviour near infinity

 

Not all curve exhibit these properties, but starting from a general list helps structure the process.

Features of a curve 1

Sketch the curve of $$ y = x^2 - 6x + 2 $$
solution - press button to display
  1. y-intercept - for a curve of the form $y = ax^2 + bx + c$, the intersect is $y=c$, in this case the intersect is $y=2$
  2. gradient near origin - for a curve of the form y = ax^2 + bx + c, the gradient at the origin is $b$, in this case, $-2$
  3. roots - for a quadratic, the roots are given by the quadratic formula, in this case, $x = 3\pm\sqrt{7}$
  4. turning points - for a quadratic, we can find the turning point by completing the square, $y = (x-3)^2 - 7$, giving a turning point go $(3,-7)$
  5. asymptotes - a quadratic has no asymptotes, so this doesn't apply
  6. behaviour near infinity - the curve looks more like $x^2$ as $x\rightarrow \pm \infty$ - U shape.
curve sketch

Features of a curve 2

Sketch the quadratic curve given by the equation below, highlighting all of its key features. $$y = -2x^2 - 12x + 18$$
solution - press button to display
  1. y-intercept - for a curve of the form $y = ax^2 + bx + c$, the intersect is $y=c$, in this case the intersect is $y=18$
  2. gradient near origin - for a curve of the form y = ax^2 + bx + c, the gradient at the origin is $b$, in this case, $-12$
  3. roots - for a quadratic, the roots are given by the quadratic formula, in this case, $x = -3\pm 3\sqrt{2}$
  4. turning points - for a quadratic, we can find the turning point by completing the square, $y = -2(x+3)^2 + 36$, giving a turning point go $(-3,36)$
  5. asymptotes - a quadratic has no asymptotes, so this doesn't apply
  6. behaviour near infinity - the curve looks more like $-2x^2$ as $x\rightarrow \pm \infty$ - $\cap$- shape.
curve sketch 2