SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Functions, 2ADV EQ-Bank 09

Using the discriminant, or otherwise, justify why the graph of  \(f(x)=-x^2+2 x-2\)  lies entirely below the \(x\)-axis.   (2 marks)

Show Answers Only

\(\Delta=b^2-4 a c=2^2-4(-1)(-2)=-4\)

\(\text{Since \(\ \Delta<0, \ y=-x^2+2 x-2\ \) does not intersect the \(x\)-axis.}\)

\(\text{Since \(\ a=-1<0, f(x)\) is an upside down parabola.}\)

\(\Rightarrow f(x)\ \text{must lie entirely below} \ x\text{-axis.}\)

Show Worked Solution

\(\Delta=b^2-4 a c=2^2-4(-1)(-2)=-4\)

\(\text{Since \(\ \Delta<0, \ y=-x^2+2 x-2\ \) does not intersect the \(x\)-axis.}\)

\(\text{Since \(\ a=-1<0, f(x)\) is an upside down parabola.}\)

\(\Rightarrow f(x)\ \text{must lie entirely below} \ x\text{-axis.}\)

Functions, 2ADV F1 EQ-Bank 1 MC

The graph of a quadratic function  \(f(x)=a x^2+b x+c\)  is drawn below.
 

Which of the following are true?

  1. \(a<0, c=0\)  and  \(b^2-4 a c=0\)
  2. \(a>0, c=0\)  and  \(b^2-4 a c=0\)
  3. \(a>0, c>0\)  and  \(b^2-4 a c>0\)
  4. \(a<0, c>0\)  and  \(b^2-4 a c=0\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Quadratic touches } x \text{-axis once only} \ \ \Rightarrow b^2-4 a c=0\ \ \text{(eliminate C)}\)

\(\text{Quadratic is inverted} \Rightarrow a<0 \ \ \text{(eliminate B)}\)

\(\text{If} \ \ c=0, f(x)=a x^2+b x+0=x(a x+b) \Rightarrow \text{cuts twice (Eliminate A)}\)

\(\Rightarrow D\)

Functions, 2ADV F1 EQ-Bank 20

Find the value of \(k\)  if  \(4kx^2-(3-4k) x+k=0\)  has one root.   (2 marks)

Show Answers Only

\(k=\dfrac{3}{8}\)

Show Worked Solution

\(4kx^2-(3-4k) x+k=0\)

\(\text{1 root}\ \Rightarrow \Delta=0\)

  \(\Delta\) \(=b^2-4 a c\)
  \(0\) \(=\left[ -\left( 3-4k \right)\right]^2-4\times 4k \times k\)
  \(0\) \(=9-24k + 16k^2-16k^2\)
  \(24k\) \(=9\)
  \(k\) \(=\dfrac{9}{24}=\dfrac{3}{8}\)

Functions, 2ADV F1 EQ-Bank 17

The tangent to the parabola  \(y=x^2+2 x-4\)  is  \(y=px-5\)  where  \(p>0\).

Find the value of \(p\).   (2 marks)

Show Answers Only

\(p=4\)

Show Worked Solution

\(\text{Intersection occurs when:}\)

\(x^2+2x-4\) \(=px-5\)  
\(x^2+(2-p)x+1\) \(=0\)  

 
\(\text{Tangent touches once}\ \Rightarrow\ \text{Discriminant}\ \Delta=0\)

\((2-p)^2-4 \times 1 \times 1\) \(=0\)  
\(4-4p+p^2-4\) \(=0\)  
\(p(p-4)\) \(=0\)  
\(p\) \(=4\ \ \ (p\gt 0)\)  
COMMENT: Key is to recognise this is a discriminant question, not a calculus application.

Functions, 2ADV F1 SM-Bank 25

Show that the parabola  \(2x^2-kx+k-2\)  has at least one real root.  (3 marks)

Show Answers Only

\(2x^2-kx+k-2=0\)

\(\Delta=b^2-4ac=(-k)^2-4 \times 2(k-2) = k^{2}-8k+16\)

\(\text{Real roots:}\ \ \Delta \geqslant 0\)

\(k^2-8k+16\) \(\geqslant 0\)  
\((x-4)^2\) \(\geqslant 0\)  

 
\(\therefore\ \text{At least one root exists for all}\ k\)

Show Worked Solution

\(2x^2-kx+k-2=0\)

\(\Delta=b^2-4ac=(-k)^2-4 \times 2(k-2) = k^{2}-8k+16\)

\(\text{Real roots:}\ \ \Delta \geqslant 0\)

\(k^2-8k+16\) \(\geqslant 0\)  
\((x-4)^2\) \(\geqslant 0\)  

 
\(\therefore\ \text{At least one root exists for all}\ k\)

Functions, 2ADV F1 SM-Bank 23

Find the values of `k` for which the expression  `x^2-3x + (4-2k)`  is always positive.  (3 marks)

Show Answers Only

`k < 7/8`

Show Worked Solution

`x^2-3x + (4-2k) > 0`

`x^2-3x + (4-2k) = 0\ \ text(is a concave up parabola)`

`=>\ text{Always positive (no roots) if}\ \ Delta < 0`
 

`b^2-4ac < 0`

`(−3)^2-4 · 1 · (4-2k)` `< 0`
`9-16 + 8k` `< 0`
`8k` `< 7`
`k` `< 7/8`