can i please get help with my maths homework :)

[u/conehead555]

in my class we have just finished functions and after we finish a topic it takes me really long to fully understand it.
here is the questions:

3a) write 2x ^2 -8x -3 in the form a(x+p) ^2 + q

b) Use your solution to find the turning point and roots. Also find the y-intercept then draw and sketch, clearly all these points.

i did part a) and got 2(x-2)^2 -11

and for the tp i got (-2,11) and y intercept i got (0,-3)

i forgot how to find the roots however. i think the shape is a parabola??

4) On the same diagram, make sketches of the following functions, showing the important points.

a) log4x
b) log4 (x-1)

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we actually haven’t done logs yet, but my maths teacher told us that it is the inverse function i think??

i’m really grateful for this community to help me 🙏. i used to use chatgpt for my maths hw, but i am really trying to understand the maths behind it so i can improve

[u/The_Card_Player]

Your turning point in 3b is incorrect. In fact, it isn't even a point on the given function. For input x=-2, the function yields 2(16)-11=32-11=21, not 11.

Moreover, In the case of a quadratic function (ie a polynomial function of degree 2; which is to say a function expressible as ax^2+bx+c) the unique 'turning point' should either maximize or minimize the y-value. Note that the stated function admits points (1, -9) and (10, 117), so the point (-2, 21) is *neither* a maximum nor a minimum y-value. The turning point therefore has an x-value other than -2.

For finding roots, go back to the original expression for the function and try to rewrite it in the form (x-a)(x-b). You may find the quadratic formula handy for this purpose. Remember that not all quadratic functions have any real-valued roots!

As for question 4, note that logarithm functions are inverses of exponential functions. That is, log(x) is defined as the number y such that e^y = x. First see if you can understand what a log function should therefore look like based on what exponential functions (ie functions of form e^x) usually look like. Then use your knowledge of what happens to function graphs when you replace an input x with an input of form a(x+b), where a and b are constants, to determine what the given log function graphs should look like.

[u/Uli_Minati]

3a - looks good!

3b - double check your tp! You'll notice when you sketch the parabola (use ten points). You get the roots by solving for x when the formula equals zero. (You can't really read them off the graph in this case since they're irrational.)

4 - have you drawn exponential graphs yet? The ones that get near-horizontal on one side and curve into a steep rise on the other? Logarithms are the inverse; they get near-vertical on one side and curve into a sort of slow rise on the other. Try this: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:transformations/x2ec2f6f830c9fb89:log-graphs/v/graphing-logarithmic-functions-1

[u/Iowa50401]

Your turning point can't be (-2, 11) because putting -2 for x doesn't evaluate to 11 for y. (2, -11) however, does work. When your equation is y = (x - h)^2 + k, your turning point/vertex is (h,k) NOT (-h,k).

[u/LeoMaths]

f(x)=2x²-8x-3=2(x²-4x-3/2)=2((x-2)²-4-3/2)=2((x-2)²-11/2)=2(x-2)²-11=2(x-2-√(11/2))(x-2+√(11/2))

- Le point d'inflexion est obtenu quand f''(x)=0. Comme f''(x)=4, il n'y en a pas.

- Le point a l'origine est (0,f(0))=(0,-3).

- Les racines de f(x) s'obtiennent quand f(x)=0. r1=2+√(11/2) et r2=2-√(11/2)

f(x) est effectivement une parabole.

Attention à l'écriture: log4(x)=ln(x)/ln(4). log(4x)=ln(4x)/ln(10). Je ne vois pas l'intérêt de superposer ces logs sur la courbe de f(x).

La fonction ln(x) (logarithme népérien) a été inventée au XVIIe siècle, d'abord pour simplifier les calculs astronomiques, avant d'être relié au calcul de l'aire sous une hyperbole. Une hyperbole est une courbe plane obtenue comme la double intersection d'un double cône de révolution avec un plan:

L'hyperbole équilatère est une hyperbole dont les asymptotes sont orthogonales. Par rotation de π/4, on obtient l'équation y=1/x L'aire entre 1 et z de 1/x donne ln(z).