Solving Equations using Graphs Flashcards

All 6 cards in this deck

What are the steps to find the straight line to draw on a given curve to solve a different equation?

e.g. graph of $y = x^2 - 3x + 1$ is drawn; solve $x^2 - 4x = 0$

1. Write the equation to be solved with $0$ on one side: $x^2 - 4x = 0$.
2. Subtract it from the curve's equation: $(x^2 - 3x + 1) - (x^2 - 4x) = x + 1$.
3. Draw $y = x + 1$ and read off the $x$-coordinates of the intersections.

When a straight line is drawn on a curve to solve an equation, which coordinates of the intersection points are the solutions?

The $x$-coordinates only; the $y$-values are ignored.

True or false? If the line needed to solve an equation is $y = 3 - \frac{x}{2}$, drawing $y = \frac{x}{2} - 3$ instead would still be correct.

False. The reversed/negative form is a different line; only $y = 3 - \frac{x}{2}$ (or a rearrangement of it such as $2y = 6 - x$) is accepted.

True or false? At a maximum point on a curve the gradient is zero, so the tangent there is parallel to the $x$-axis.

True. The curve is momentarily flat at a maximum point.

True or false? A curve that touches the $x$-axis at its minimum point $(a, 0)$ and does not meet the $x$-axis anywhere else has three different values of $x$ for which $y = 0$.

False. Touching the axis there gives only one value of $x$ with $y = 0$, namely $x = a$.

What does it mean to say a function is increasing over a section of its graph?

e.g. between a minimum point and the next maximum point

$y$ gets larger as $x$ gets larger — the curve goes upwards from left to right.

e.g. between a minimum and the following maximum the function is increasing