Algebra Revision Notes

Understanding Graphical Solutions

Solving an equation like f(x) = g(x) means finding the values of x for which f(x) is equal to g(x). Graphically, this corresponds to finding the x-coordinates of the points where the graph of y = f(x) intersects the graph of y = g(x). The y-values at these intersection points will be equal for both functions.

For example, to solve x² - 2x = 3, you can plot y = x² - 2x and y = 3. The x-coordinates where these two graphs meet are the solutions. The solutions will be approximate because drawing and reading from a graph introduces some inaccuracy.

Steps for Finding Approximate Solutions

1. Identify the two functions: If your equation is f(x) = g(x), you will plot y = f(x) and y = g(x). If it's f(x) = 0, you plot y = f(x) and find where it intersects the x-axis (y=0).
2. Plot the graphs accurately: Create a table of values for each function over a relevant range of x. Plot the points carefully and draw a smooth curve (or straight line) through them. Use a sharp pencil and ruler for straight lines.
3. Locate intersection points: Identify all points where the two graphs cross or touch each each other. These are the points of intersection.
4. Read the x-coordinates: For each intersection point, read the corresponding x-coordinate from the x-axis. These x-coordinates are your approximate solutions to the equation.
5. Estimate accurately: When reading off values, always try to estimate to one decimal place, or to the precision indicated by the scale of the axes, unless the question specifies a different degree of accuracy.

Interpreting Results

The number of intersection points corresponds to the number of real solutions to the equation. If the graphs don't intersect, there are no real solutions. If they touch at one point (tangent), there is one real solution. If they cross at multiple points, there are multiple real solutions.

Always ensure that your graph is drawn clearly enough to allow for accurate reading. Slight errors in plotting can lead to different approximate solutions. The term 'approximate' is key here; graphical methods rarely give exact solutions unless the intersection points fall precisely on grid lines.

Common Scenarios

A frequent application is solving a quadratic equation ax² + bx + c = 0. You can plot y = ax² + bx + c and find the x-intercepts. Alternatively, you might be asked to solve an equation like x² - 5x + 2 = 3x - 1. Here, you would plot y = x² - 5x + 2 and y = 3x - 1, then find their intersections. Sometimes you need to rearrange the given equation into a form that matches one of the plotted graphs and an additional linear graph.

Core method

A graph can provide approximate solutions where curves meet or where a curve crosses a chosen horizontal level. To solve f(x) = 0, read the x-coordinates of x-axis intersections. To solve f(x) = g(x), draw or compare both graphs and read the x-coordinates of their intersections. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

Use a sensible scale, plot accurately and draw a smooth curve where appropriate. State solutions to a precision supported by the graph. For y = x² and y = 5, the intersections occur at approximately x = −2.24 and x = 2.24; a coarse graph might justify only x ≈ −2.2 and 2.2. Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.

Understanding Graphical Solutions

Solutions to an equation can be found graphically by identifying the points where the graphs of related functions intersect. If you have an equation of the form $f(x) = g(x)$, you can plot $y = f(x)$ and $y = g(x)$. The x-coordinates of the intersection points are the solutions to the equation. Alternatively, for an equation $h(x) = 0$, you can plot $y = h(x)$ and find where it crosses the x-axis (where $y=0$). These x-intercepts are the solutions.

Steps for Approximating Solutions Graphically

1. Rearrange the Equation (if necessary): Sometimes it's easier to plot simpler functions. For example, $x^2 - 2x - 3 = 0$ can be solved by plotting $y = x^2 - 2x - 3$ and finding x-intercepts. Or, by rearranging to $x^2 = 2x + 3$, you can plot the curves $y = x^2$ and $y = 2x + 3$. The latter often involves a straight line, which is easier to draw.
2. Draw the Graphs: Create a table of values for each function and plot the points accurately on graph paper. Ensure axes are correctly labelled and scaled.
3. Identify Intersection Points: Locate where the plotted graphs cross. These are the points where the 'y' values are equal, and therefore the 'x' values are the solutions.
4. Read Off Approximate Solutions: Carefully read the x-coordinates of the intersection points from the graph. The precision depends on the scale of your graph and how accurately you can interpret the coordinates.

Accuracy and Interpretation

The accuracy of graphical solutions is limited by the precision of your drawing and reading. Always use a sharp pencil and a ruler for straight lines. When estimating a value between grid lines, try to be as precise as possible. Questions often specify the required degree of accuracy (e.g., to one decimal place). If a solution seems to be at an integer value, double-check it. Bear in mind that these are *approximations*, not exact solutions unless explicitly stated or obvious from the graph geometry.

Solving Complex Equations

Graphical methods are particularly useful for equations involving higher powers or transcendental functions (e.g., trigonometric, exponential, logarithmic) that are difficult to solve algebraically. The principle remains the same: define two functions whose intersection points represent the solutions to the original equation, then plot and read from the graph.

Understanding Iteration

Iteration is a process of repeatedly applying a formula to get closer to a solution. For an equation f(x) = 0, we rearrange it into the form x = g(x). We start with an initial guess, x₀, and then generate a sequence of approximations using the iterative formula xₙ₁₁ = g(xₙ). Each new value is calculated using the previous one. This method helps find approximate solutions for equations that are difficult to solve directly. The accuracy of the approximation improves with each step, provided the iteration converges.

Forming an Iterative Formula

To use iteration, the first step is to rearrange the given equation f(x) = 0 into an iterative form x = g(x). There can be multiple ways to do this. For example, if you have x² - 3x + 1 = 0, you could rearrange it to x = (x² + 1)/3 or x = √(3x - 1). The choice of rearrangement is crucial because it affects whether the iteration will converge or diverge. Sometimes the question will provide the iterative formula directly, or guide you on how to form it.

Convergence and Divergence

An iterative process 'converges' if the sequence of approximations gets closer and closer to the actual root. It 'diverges' if the approximations move further away. Graphically, convergence often looks like a 'cobweb' diagram spiralling inwards or a 'staircase' diagram stepping towards the root. The condition for local convergence is that |g'(x)| < 1 at the root. If this condition is not met, the iteration may diverge or oscillate without settling on a single value. When performing calculations, you'll observe convergence as successive values becoming very close or identical to a certain decimal place.

Performing Iterations and Stopping Conditions

Start with a given initial value x₀. Calculate the next value x₁ using the formula x₁ = g(x₀). Then use x₁ to find x₂ = g(x₁), and so on. Continue this process until the values converge to the required degree of accuracy. For example, if asked for an answer to 3 decimal places, you would continue iterating until two successive values round to the same number when rounded to 3 decimal places. It's good practice to show at least three iterations in your working, even if you find the answer quicker.