Equation from curve features questions with answers

Questions

1. Question 1From the x-intercepts

   The diagram shows the curve $y = (x - a)(x - b)$. Find the equation of the curve.

   Show the worked solution

   1. Step 1: Read the x-interceptsThe curve meets the $x$-axis at $x = -3$ and $x = 2$.
   2. Step 2: Each x-intercept gives a bracket: x = a gives (x − a)$a = -3,\ b = 2$
   3. Step 3: U-shaped: no minus in front$y = (x - 2)(x + 3)$
   4. Step 4: Check with the y-intercept$x = 0$: $y = (-2)(3)$$y = -6$, as on the diagram.
2. Question 2x-intercepts, ∩-shaped

   The diagram shows the curve $y = -(x - a)(x - b)$. Find the equation of the curve.

   Show the worked solution

   1. Step 1: Read the x-interceptsThe curve meets the $x$-axis at $x = -5$ and $x = -2$.
   2. Step 2: Each x-intercept gives a bracket: x = a gives (x − a)$a = -5,\ b = -2$
   3. Step 3: ∩-shaped: a minus in front$y = -(x + 2)(x + 5)$
   4. Step 4: Check with the y-intercept$x = 0$: $y = -(2)(5)$$y = -10$, as on the diagram.
3. Question 3From a minimum point

   The diagram shows the curve $y = (x - h)^2 + k$. Find the equation of the curve.

   Show the worked solution

   1. Step 1: Read the minimum pointTurning point: $x = -2$, $y = 3$
   2. Step 2: The turning point (h, k) goes straight into the form$h = -2,\ k = 3$
   3. Step 3: A minimum: no minus in front$y = (x + 2)^2 + 3$
   4. Step 4: Check with the y-intercept$x = 0$: $y = (2)^2 + 3$$y = 7$, as on the diagram.
4. Question 4From a maximum point

   The diagram shows the curve $y = -(x - h)^2 + k$. Find the equation of the curve.

   Show the worked solution

   1. Step 1: Read the maximum pointTurning point: $x = -3$, $y = -1$
   2. Step 2: The turning point (h, k) goes straight into the form$h = -3,\ k = -1$
   3. Step 3: A maximum: a minus in front$y = -(x + 3)^2 - 1$
   4. Step 4: Check with the y-intercept$x = 0$: $y = -(3)^2 - 1$$y = -10$, as on the diagram.
5. Question 5From the line of symmetry and two points: find a, b and c

   The curve $y = ax^2 + bx + c$ passes through the points $(0, -6)$ and $(-4, 26)$. Its line of symmetry is $x = 2$. Find the values of $a$, $b$ and $c$, and write down the equation of the curve.

   Show the worked solution

   1. Step 1: The curve passes through (0, c): put x = 0$c = -6$
   2. Step 2: (0, c) is 2 units from the line of symmetry x = 2: its mirror image is the same distance on the other side, at the same heightMirror image: $(4, -6)$
   3. Step 3: Put the mirror image into y = ax² + bx + c: the c's cancel, giving b in terms of a$-6 = a(4)^2 + b(4) + -6$$0 = 16a + 4b$$b = -4a$
   4. Step 4: Put (-4, 26) into the equation, with b = -4a$26 = a(-4)^2 + (-4a)(-4) + -6$$32a = 32$$a = 1$
   5. Step 5: Then b$b = -4$
   6. Step 6: Write the equation$y = x^{2} - 4x - 6$