IB Maths AA HL Topic 2: Functions questions

Write down the $y$-intercept of $\ell$ and determine its gradient.

The line $m$ is parallel to $\ell$ and passes through $P(2,1)$ Determine the equation of $m$ in the form $y=mx+c$

The line $n$ is perpendicular to $\ell$ and passes through the point where $m$ intersects the $y$-axis. Determine the equation of $n$ in the form $Ax+By+C=0$ where $A,B,C$ are integers with no common factor and $A>0$

At $t=0$ state which analyst predicts the greater annual demand.

Sketch the two models on the same axes for $0\leq t\leq15$ Use your sketch to estimate, to the nearest year, when the forecasts agree.

Determine analytically the exact times at which the two forecasts agree. Give your answers in years.

Determine the values of $p$ for which the equation has (i) two distinct real roots, (ii) two equal real roots, and (iii) no real roots.

Hence, when $p=3$ solve

State the equations of the vertical and horizontal asymptotes of the graph of $g$

Sketch the graph of $g$ showing and labelling its asymptotes and its intercepts with the axes.

Determine $g^{-1}(x)$ including its domain, and hence state whether $g$ is self-inverse.

Determine the equations of the vertical and horizontal asymptotes of the graph of $g$

Verify algebraically that $g(g(x))=x$ for $x\ne 1$ Hence state the line of symmetry of the graph of $g$ and describe how reflection in this line maps the asymptotes found in part (a).

Determine the value of $a$ correct to three significant figures.

Let $T(V)$ give the time, in seconds, at which the voltage is $V$ Determine $T(V)$ stating its domain.

Hence determine the time at which the model predicts the voltage is 2.5 V. State whether the model predicts that the voltage reaches 0 V at any finite time, and justify your answer.

Write down $F(x)$

The graph $y=F(x)$ is reflected in the $x$-axis and then translated upwards by $3$ units, giving $y=g(x)$ Determine $g(x)$ in completed-square form and state the coordinates of its turning point.

Instead, starting from $y=F(x)$ translate it upwards by $3$ units and then reflect it in the $x$-axis, giving $y=h(x)$ Hence, or otherwise, determine whether $h$ has any $x$-intercepts. Give a reason.

Write down $f(-2)$

On the same axes as Figure 1, sketch the graph of $y=g(x)$ Mark the image of the point on $y=f(x)$ for which $x=-2$

Using Figure 1, determine all values of $x$ for which $g(x)=3$

For the point $(-2,2)$ on $y=f(x)$ determine its image ordinate if the translation upwards by $1$ unit is performed after the reflection but before the vertical stretch. Hence state whether this order of transformations gives the same graph as $y=g(x)$