Transformations of quadratic graphs - Completing the square and sketching

4 marks Easy -1.3

9

A curve has equation $y = x ^ { 2 } - 4$. Find the $x$-coordinates of the points on the curve where $y = 21$.
The curve $y = x ^ { 2 } - 4$ is translated by $\binom { 2 } { 0 }$. Write down an equation for the translated curve. You need not simplify your answer.

4 marks Easy -1.2

1

A curve has equation $y = x ^ { 2 } - 4$. Find the $x$-coordinates of the points on the curve where $y = 21$.
The curve $y = x ^ { 2 } - 4$ is translated by $\binom { 2 } { 0 }$. Write down an equation for the translated curve. You need not simplify your answer.

6 marks Moderate -0.8

4 The quadratic function $\mathrm { f } ( x )$ is given by $\mathrm { f } ( x ) = x ^ { 2 } - 3 x + 2$.

Write $\mathrm { f } ( x )$ in the form $( \mathrm { x } + \mathrm { a } ) ^ { 2 } + \mathrm { b }$, where $a$ and $b$ are constants.
Write down the coordinates of the minimum point on the graph of $y = f ( x )$.
Describe fully the transformation that maps the graph of $y = f ( x )$ onto the graph of $y = ( x + 1 ) ^ { 2 } - \frac { 1 } { 4 }$.

5 marks Easy -1.3

1

Express $x ^ { 2 } + 8 x + 2$ in the form $( x + a ) ^ { 2 } + b$.
Write down the coordinates of the turning point of the curve $y = x ^ { 2 } + 8 x + 2$.
State the transformation(s) which map(s) the curve $y = x ^ { 2 }$ onto the curve $y = x ^ { 2 } + 8 x + 2$.

11 marks Moderate -0.8

5

Express $( x - 5 ) ( x - 3 ) + 2$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
(3 marks)
1. Sketch the graph of $y = ( x - 5 ) ( x - 3 ) + 2$, stating the coordinates of the minimum point and the point where the graph crosses the $y$-axis.
2. Write down an equation of the tangent to the graph of $y = ( x - 5 ) ( x - 3 ) + 2$ at its vertex.
Describe the geometrical transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = ( x - 5 ) ( x - 3 ) + 2$.

10 marks Moderate -0.8

2

Express $x ^ { 2 } + 8 x + 19$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
Hence, or otherwise, show that the equation $x ^ { 2 } + 8 x + 19 = 0$ has no real solutions.
Sketch the graph of $y = x ^ { 2 } + 8 x + 19$, stating the coordinates of the minimum point and the point where the graph crosses the $y$-axis.
Describe geometrically the transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = x ^ { 2 } + 8 x + 19$.

12 marks Easy -1.2

4

Express $x ^ { 2 } + 5 x + 7$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are rational numbers.
(3 marks)
A curve has equation $y = x ^ { 2 } + 5 x + 7$.
1. Find the coordinates of the vertex of the curve.
2. State the equation of the line of symmetry of the curve.
3. Sketch the curve, stating the value of the intercept on the $y$-axis.
Describe the geometrical transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = x ^ { 2 } + 5 x + 7$.

8 marks Moderate -0.8

5

Express $x ^ { 2 } + 3 x + 2$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are rational numbers.
A curve has equation $y = x ^ { 2 } + 3 x + 2$.
1. Use the result from part (a) to write down the coordinates of the vertex of the curve.
2. State the equation of the line of symmetry of the curve.
The curve with equation $y = x ^ { 2 } + 3 x + 2$ is translated by the vector $\left[ \begin{array} { l } 2 \\ 4 \end{array} \right]$. Find the equation of the resulting curve in the form $y = x ^ { 2 } + b x + c$.

8 marks Moderate -0.3

$$f(x) = x^2 - 10x + 17.$$

Express $f(x)$ in the form $a(x + b)^2 + c$. [3]
State the coordinates of the minimum point of the curve $y = f(x)$. [1]
Deduce the coordinates of the minimum point of each of the following curves:
1. $y = f(x) + 4$, [2]
2. $y = f(2x)$. [2]

6 marks Moderate -0.8

The curve $C_1$ has equation $y = 2x^2 - 20x + 42$ Express the equation of $C_1$ in the form $$y = a(x - h)^2 + c$$ where $a$, $b$ and $c$ are integers. [3 marks]
Write down the coordinates of the minimum point of $C_1$ [1 mark]
The curve $C_1$ is mapped onto the curve $C_2$ by a stretch in the $y$-direction. The minimum point of $C_2$ is at $(5, -4)$ Find the equation of $C_2$ [2 marks]