Coordinate Geometry and Quadratic Graphs PDF

1. (a) The diagram below shows the graphs of two functions on the same pair of axes. The lines g and h are perpendicular. - Determine the: - (i) equation that represents the line g - (ii) equation that represents the line h - (iii) coordinates of the point P. Show all wor...

1. (a) The diagram below shows the graphs of two functions on the same pair of axes. The lines g and h are perpendicular. - Determine the: - (i) equation that represents the line g - (ii) equation that represents the line h - (iii) coordinates of the point P. Show all working. (b) (i) Write $4x^2 - 24x + 31$ in the form $a(x + h)^2 + k$. - (ii) On the axes below, sketch the graph of $4x^2 - 24x + 31$, indicating the coordinates of the maximum/minimum point and the y-intercept. - (iii) State the equation of the axis of symmetry. 2. The graph below represents the function $f(x) = x^2 -3x - 3$. - Use the graph to determine: - (a) the value of $f(x)$ when $x = 2$. - (b) the value of $f(x)$ when $x = 1.5$ - (c) the values of $x$ for which $f(x) = 0$ - (d) the minimum value of $f(x)$ - (e) the value of $x$ at which $f(x)$ is a minimum. - (f) the solution of $x^2 - 3x - 3 = 5$ - (g) the interval on the domain for which $f(x)$ is less than $-3$. 3. An answer sheet is provided for this question. - The graph of the quadratic function $y = x^2$ for $-4 \le x \le 4$ - The coordinates of the points M and N are $(-1, y)$ and $(x, 9)$ respectively. - Determine the value of: - (i) $x$ - (ii) $y$ - Determine: - (i) the gradient of the line MN - (ii) the equation of the line MN - (iii) the equation of the line parallel to MN, and passing through the origin. - On the answer sheet provided, carefully draw the tangent line to the graph $y = x^2$ at the point (2, 4). - Estimate the gradient of the tangent to the curve at (2, 4). 4. (a) Complete the table for the function $y = -x^2 + x + 7$. | x | y | |:--:|:--:| | -3 | 1 | | -2 | 1 | | -1 | 7 | | 0 | 5 | | 1 | -5 | | 2 | | | 3 | | | 4 | | - (b) Using a scale of 2cm to represent 1 unit on the $x$-axis and 1 cm to represent 1 unit on the $y$-axis, draw the graph of $y = -x^2 + x + 7$ for $-3 < x \le 4$. - (c) Write down the coordinates of the maximum/minimum point of the graph. - (d) Write down the equation of the axis of symmetry of the graph. - (e) Use your graph to find the solutions of the equation $-x^2 + x + 7 = 0$. - (f) On the grid on page 24, draw a line through the points $(-3, -1)$ and $(0, 8)$. - Determine the equation of this line in the form $y = mx + c$. 5. A graph sheet is provided for this question. The table below is designed to show values of $x$ and $y$ for the function $y = x^2 - 2x - 3$ for integer values of $x$ from $-2$ to $4$. | x | y | |:--:|:--:| | -2 | 5 | | -1 | | | 0 | -3 | | 1 | -4 | | 2 | -3 | | 3 | 5 | | 4 | | - (a) Complete the table for the function $y = x^2 - 2x - 3$. - (b) On the graph on page 13, plot the graph of $y=x^2-2x-3$ using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis. - (c) On the graph on page 13, draw a smooth curve passing through the points on your graph. - (d) Complete the following sentences using information from your graph. - (i) The values of $x$ for which $x^2 - 2x - 3 = 0$ are ___ and ___. - (ii) The minimum value of $x^2 - 2x - 3$ is ___. - (iii) The equation of the line of symmetry of the graph of $y = x^2 – 2x – 3$ is ___.