Graphs of Functions Flashcards

All 31 cards in this deck

What shape is the graph of a quadratic function?

e.g. $y = x^2 - 3x + 1$

A parabola (a symmetrical U-shape).
The $x^2$ term is positive, so this one opens upwards.

What shape is the graph of a simple cubic function such as $y = x^3$?

An S-shaped curve through the origin, rising from bottom-left to top-right.

What does the graph of a reciprocal function $y = \frac{a}{x}$ look like?

e.g. $y = \frac{4}{x}$

Two separate curved branches in opposite quadrants, with both axes as asymptotes.
For $y = \frac{4}{x}$ the branches lie in the top-right and bottom-left quadrants.

State the key features of the graph of $y = k^x$ for $k > 1$.

e.g. $y = 2^x$

Passes through $(0,1)$, always positive, increasing ever more steeply, with the $x$-axis as an asymptote.

True or false? The graph of $y = 3^x$ passes through the origin.

False.
It passes through $(0,1)$, because $3^0 = 1$, and never touches the $x$-axis.

What does the graph of $y = k^x$ look like when $0 < k < 1$?

e.g. $y = 0.5^x$

A decreasing curve through $(0,1)$, getting closer and closer to the $x$-axis as $x$ increases.

What are the roots of a quadratic graph?

The $x$-coordinates of the points where the curve crosses the $x$-axis, i.e. the solutions of $y = 0$.

How do you find the $y$-intercept of a quadratic from its equation?

e.g. $y = x^2 - 6x + 4$

It is the constant term (the value of $y$ when $x = 0$).
Here the $y$-intercept is $4$, i.e. $(0,4)$.

What is the turning point of a quadratic graph?

The maximum or minimum point, where the curve changes direction; it lies on the line of symmetry.

What is the equation of the line of symmetry of a parabola with turning point $(2,-13)$?

$x = 2$.
The line of symmetry is $x =$ the $x$-coordinate of the turning point.

True or false? Every quadratic graph crosses the $x$-axis twice.

False.
A parabola can cross the $x$-axis twice, touch it once, or not meet it at all.

How can you tell from its equation whether a parabola opens upwards or downwards?

Positive coefficient of $x^2$ opens upwards (minimum); negative coefficient opens downwards (maximum).

What are the steps to draw a graph from a table of values?

e.g. $y = 5 - x^3$ for $-2 \le x \le 2$

1. Substitute each $x$ into the equation: $x = -2$ gives $y = 13$
2. Plot the points: $(-2,13), (-1,6), (0,5), (1,4), (2,-3)$
3. Join them with a single smooth curve.

How should plotted points of a quadratic or cubic graph be joined?

With a smooth curve (freehand is fine) — not with ruled straight line segments.

True or false? When $x = -2$, the value of $x^2$ is $-4$.

False.
$(-2)^2 = 4$; squaring a negative number gives a positive result.

What symmetry should you see in a completed table of values for a quadratic?

The $y$-values repeat symmetrically either side of the turning point.
e.g. $\ldots, -3, -4, -3, \ldots$

Why can a table of values for $y = \frac{a}{x}$ never include $x = 0$?

Because dividing by $0$ is undefined, so the graph has no point there (the $y$-axis is an asymptote).

What are the steps to solve a related equation by drawing a straight line on a given curve?

e.g. use the graph of $y = x^2 + x - 4$ to solve $x^2 + x - 6 = 0$

1. Rearrange so one side matches the curve: $x^2 + x - 4 = 2$
2. Draw the line $y = 2$ on the same axes
3. Read the $x$-coordinates where the line meets the curve.

How do you use a drawn curve to estimate the value of $f(a)$?

e.g. estimate $f(1.5)$

Go up from $x = 1.5$ on the $x$-axis to the curve, then straight across to the $y$-axis and read off the value.

True or false? Solutions found from a graph should be written as coordinates.

False.
Give the $x$-values only, e.g. $0.8$ and $5.2$.

True or false? If the line $y = k$ does not meet the curve $y = f(x)$, then $f(x) = k$ has no solutions in that range.

True.
Solutions correspond to points of intersection of the line and the curve.

State the key features of the graph of $y = \sin x^\circ$.

A wave through $(0,0)$, repeating every $360^\circ$, with maximum $1$ and minimum $-1$.

State the key features of the graph of $y = \cos x^\circ$.

A wave starting at $(0,1)$, repeating every $360^\circ$, with maximum $1$ and minimum $-1$.

State the key features of the graph of $y = \tan x^\circ$.

Period $180^\circ$, crossing the $x$-axis at $0^\circ, 180^\circ, 360^\circ, \ldots$, with vertical asymptotes at $90^\circ, 270^\circ, \ldots$

For $0 \le x \le 360$, at which values of $x$ does the graph of $y = \sin x^\circ$ cross the $x$-axis?

$x = 0$, $180$ and $360$.

True or false? The maximum value of $\tan x^\circ$ is $1$.

False.
$\tan x^\circ$ has no maximum or minimum — it increases without limit towards each asymptote.

True or false? $\sin x^\circ$ is negative for all $x$ between $180$ and $360$.

True.
The sine curve lies below the $x$-axis between $180^\circ$ and $360^\circ$.

What are the steps to solve $\sin x^\circ = k$ for $0 \le x \le 360$?

e.g. $2\sin x^\circ = -1$

1. Rearrange: $\sin x^\circ = -0.5$
2. Find the related acute angle: $\sin^{-1}0.5 = 30$
3. Use the sine graph's symmetry (sine negative after $180^\circ$): $x = 210$ and $x = 330$.

If $x$ is a solution of $\cos x^\circ = k$, how do you find the other solution in $0 \le x \le 360$?

Use $360 - x$.
e.g. if $x = 60$ then the other solution is $300$.

If $x$ is a solution of $\tan x^\circ = k$, how do you find further solutions?

Add or subtract $180$ each time, since the tangent graph has period $180^\circ$.

True or false? $\sin x^\circ = 0.6$ has exactly one solution for $0 \le x \le 360$.

False.
It has two, symmetric about $x = 90$ (e.g. about $37$ and $143$).