How many times can a quadratic graph intersect the x -axis ?

A quadratic graph can intersect the x -axis : zero times, one time, two times.

How does factorising a quadratic help you to sketch the graph of the quadratic?

Factorising a quadratic helps you to sketch the graph of the quadratic because it gives you the x -intercepts of the graph.

How does completing the square for a quadratic help you to sketch the graph of the quadratic?

Completing the square for a quadratic helps you to sketch the graph of the quadratic because it gives you the coordinates of the turning point of the graph.

True or False? A GDC can be used to find the solutions to a quadratic equation .

True. A GDC can be used to find the solutions to a quadratic equation .

What information is given by the discriminant of a quadratic function?

The discriminant of a quadratic function identifies the number of distinct real roots that its graph has. Also, it it identifies the number of distinct real solutions of the equation formed by setting the quadratic equal to zero.

How many real solutions does a quadratic equation have if the discriminant of the quadratic function is negative ?

If the discriminant of a quadratic function is negative , then the equation has no real solutions .

How many real solutions does a quadratic equation have if the discriminant of the quadratic function is zero ?

If the discriminant of a quadratic function is zero , then the equation has one (repeated) real solution .

How many real roots does a quadratic graph have if the discriminant of the quadratic function is positive ?

If the discriminant of a quadratic function is positive , then its graph has two distinct real roots .

IBMathsDPAnalysis & Approaches (AA)HLFlashcards2. FunctionsQuadratic Functions & Graphs

Quadratic Functions & Graphs (DP IB Analysis & Approaches (AA))

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FrontQuadratic Functions
What does the graph of a quadratic function $f (x) = a x^{2} + b x + c$ look like when $a < 0$ ?
FrontQuadratic Functions
How many times can a quadratic graph intersect the x-axis?
BackQuadratic Functions
A quadratic graph can intersect the x-axis:
zero times,
one time,
two times.

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Cards in this collection (41)

What does the graph of a quadratic function $f (x) = a x^{2} + b x + c$ look like when $a < 0$ ?
The graph of a quadratic function $f (x) = a x^{2} + b x + c$ is a parabola with a maximum turning point when $a < 0$ .
True or False?
A quadratic graph of the form $f (x) = a x^{2} + b x + c$ always crosses the y-axis exactly once.
True.
A quadratic graph of the form $f (x) = a x^{2} + b x + c$ always crosses the y-axis exactly once.
How many times can a quadratic graph intersect the x-axis?
A quadratic graph can intersect the x-axis:
- zero times,
- one time,
- two times.
True or False?
$(- h, k)$ are the coordinates of the turning point of the graph of the function $f (x) = a {(x - h)}^{2} + k$ .
False.
$(h, k)$ are the coordinates of the turning point of the graph of the function $f (x) = a {(x - h)}^{2} + k$ .
You change the sign in front of the $h$ .
True or False?
If a quadratic graph crosses the x-axis at the points $(p, 0)$ and $(q, 0)$ then the equation of the line of symmetry is $x = \frac{p + q}{2}$ .
True.
If a quadratic graph crosses the x-axis at the points $(p, 0)$ and $(q, 0)$ then the equation of the line of symmetry is $x = \frac{p + q}{2}$ .
If the turning point of a quadratic graph is (-2, 5), what is an equation of the quadratic function?
If the turning point of a quadratic graph is (-2, 5), then the equation of the quadratic function looks like $f (x) = a {(x + 2)}^{2} + 5$ .
The constant $a$ is needed.
If the x-intercepts of a quadratic graph are (-2, 0) and (3, 0), what is an equation of the quadratic function?
If the x-intercepts of a quadratic graph are (-2, 0) and (3, 0), then the equation of the quadratic function looks like $f (x) = a (x + 2) (x - 3)$ .
The constant $a$ is needed.
If you know the coordinates of 3 points on a quadratic curve, how could you find the equation of the quadratic function?
If you know the coordinates of 3 points on a quadratic curve, to find the equation you could:
- substitute each pair of coordinates into the equation $y = a x^{2} + b x + c$ to form three equations with a, b, c as unknowns,
- solve the system of linear equations to find the values of a, b, and c.
True or False?
The value of $c$ in the equation of the quadratic $f (x) = a x^{2} + b x + c$ does not affect the position of the line of symmetry of the quadratic graph.
True.
The value of $c$ in the equation of the quadratic $f (x) = a x^{2} + b x + c$ does not affect the position of the line of symmetry of the quadratic graph.
The equation for the line of symmetry is $x = - \frac{b}{2 a}$ .
How does factorising a quadratic help you to sketch the graph of the quadratic?
Factorising a quadratic helps you to sketch the graph of the quadratic because it gives you the x-intercepts of the graph.
An expression $x^{2} + 8 x + 12$ factorises to $(x + p) (x + q)$ .
Explain how the numbers $p$ and $q$ relate to the numbers $8$ and $12$ .
If $x^{2} + 8 x + 12$ factorises to $(x + p) (x + q)$ , then:
- $p + q = 8$ (the numbers must add to give 8).
- $p q = 12$ (the numbers must multiply to give 12).
True or False?
If $x^{2} - 54 x + 288$ factorises to $(x + p) (x + q)$ then $p$ and $q$ must both be negative.
True.
If $x^{2} - 54 x + 288$ factorises to $(x + p) (x + q)$ , then $p$ and $q$ multiply to give 288, which is positive. That means that $p$ and $q$ could both be positive or both be negative.
But since $p$ and $q$ add to give -54 which is negative, then at least one of them is negative.
The two facts above mean that $p$ and $q$ are both negative.
When factorising the quadratic $a x^{2} + b x + c$ , you need to find two numbers that multiply to make what value?
When factorising the quadratic $a x^{2} + b x + c$ , you need to find two numbers that multiply to make the value $a c$ .
Factorise $a^{2} x^{2} - c^{2}$ .
$a^{2} x^{2} - c^{2}$ can be factorised as $(a x + c) (a x - c)$ .
This is an example of difference of two squares.
Explain how to use the difference of two squares to factorise $4 x^{2} - 25$ .
To factorise $4 x^{2} - 25$ using difference of two squares, the $4 x^{2}$ can be thought of as ${(2 x)}^{2}$ . So $4 x^{2} - 25$ is ${(2 x)}^{2} - 5^{2}$ .
The difference of two squares can then be used giving $(2 x + 5) (2 x - 5)$ .
A calculator gives the solutions to $2 x^{2} + 7 x + 3 = 0$ as $- 3$ and $- \frac{1}{2}$ .
Explain why this tells you that $2 x^{2} + 7 x + 3$ can be factorised into double brackets.
If the solutions to a quadratic equation are integers or (rational) fractions, then the quadratic factorises.
The solutions are $- 3$ and $- \frac{1}{2}$ which are integers or fractions, so it must factorise.
The factorisation is $2 x^{2} + 7 x + 3 = (2 x + 1) (x + 3)$ .
How does completing the square for a quadratic help you to sketch the graph of the quadratic?
Completing the square for a quadratic helps you to sketch the graph of the quadratic because it gives you the coordinates of the turning point of the graph.
When completing the square for the expression $x^{2} + b x + c$ , explain how to find the value of $p$ in the expression ${(x + p)}^{2} + q$ .
Completing the square for $x^{2} + b x + c$ gives the form ${(x + p)}^{2} + q$ .
The value of $p$ is half of the value of $b$ .
What is the first step to completing the square of the quadratic expression $a x^{2} + b x + c$ where $a$ is not equal to 1, e.g. $3 x^{2} + 12 x - 8$ ?
The first step to completing the square of the quadratic expression $a x^{2} + b x + c$ where $a$ is not equal to 1 is to factorise out $a$ from the terms containing $x$ .
This gives $a [x^{2} + \frac{b}{a} x] + c$ .
You can then complete the square inside the big brackets.
E.g. to complete the square of the expression $3 x^{2} + 12 x - 8$ , the first step would be to factor out 3 from the first two terms, giving $3 [x^{2} + 4 x] - 8$ .
True or False?
The coordinates of the turning points on the curves $y = {(x + 2)}^{2} + 4$ and $y = 3 {(x + 2)}^{2} + 4$ are the same.
True.
The coordinates of the turning points on the curves $y = {(x + 2)}^{2} + 4$ and $y = 3 {(x + 2)}^{2} + 4$ are the same.
The coordinates of the turning point on $y = a {(x - h)}^{2} + k$ are always $(h, k)$ , regardless of the value of $a$ (even if $a < 0$ ).
State the quadratic formula for the solutions of $a x^{2} + b x + c = 0$ .
The quadratic formula, giving the solutions of $a x^{2} + b x + c = 0$ , is $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ .
This is given in the exam formula booklet.
True or False?
A GDC can be used to find the solutions to a quadratic equation.
True.
A GDC can be used to find the solutions to a quadratic equation.
What are the solutions to a quadratic equation where the quadratic expression has been factorised in the form $a (x - p) (x - q) = 0$ ?
The solutions to a quadratic equation where the quadratic expression has been factorised in the form $a (x - p) (x - q) = 0$ are $x = p$ and $x = q$ .
These come from setting the factors $(x - p)$ and $(x - q)$ equal to zero, and then solving for $x$ .
How would you find the solutions of a factorised quadratic equation like $3 (x + 5) (2 x - 1) = 0$ ?
To find the solutions to a factorised quadratic equation, you need to set each factor equal to zero and then solve.
For example, the solutions to $3 (x + 5) (2 x - 1) = 0$ are the solutions to $x + 5 = 0$ and $2 x - 1 = 0$ .
How would you find the solutions to a quadratic equation of the form $a {(x - h)}^{2} + k = 0$
For example, how would you find the roots of the equation $2 {(x - 3)}^{2} - 5 = 0$ ?
To find the solutions to a quadratic equation of the form $a {(x - h)}^{2} + k = 0$ , you would rearrange to make $x$ the subject. Remember to include $\pm$ when taking the square root.
For example, for $2 {(x - 3)}^{2} - 5 = 0$ :
$\begin{array}{rcl} {(x - 3)}^{2} & = & \frac{5}{2} \\ x - 3 & = & \pm \sqrt{\frac{5}{2}} \\ x & = & 3 \pm \sqrt{\frac{5}{2}} \end{array}$
True or False?
When solving a quadratic equation of the form $x (a x + b) = 0$ , the first step is to divide both sides by $x$ .
False.
It is a common mistake to divide both sides by $x$ at the beginning when solving a quadratic equation of the form $x (a x + b) = 0$
It is a mistake because if you do this you will lose the $x = 0$ solution.
Instead, solve $x = 0$ and $a x + b = 0$ .
True or False?
You can change the signs of each term in an inequality. For example, If $4 - x^{2} \geq 0$ then $x^{2} - 4 \geq 0$ .
False.
You can not change the signs of each term in an inequality. If $4 - x^{2} \geq 0$ then $x^{2} - 4 \leq 0$ .
If you multiply or divide an inequality by a negative number then you need to flip the inequality sign. ('Changing the signs' is the same as multiplying by -1.)
True or False?
If $(x - 2) (x + 1) \geq 0$ then $x \geq - 1$ or $x \geq 2$ .
False.
If $(x - 2) (x + 1) \geq 0$ then $x \leq - 1$ or $x \geq 2$ . You should use a sketch to determine the inequalities.
If $a x^{2} + b x + c = 0$ has two distinct roots and $a > 0$ , then which set of values satisfies the inequality $a x^{2} + b x + c < 0$ : the values between the two roots or the values not between the roots?
If $a x^{2} + b x + c = 0$ has two distinct roots and $a > 0$ , then the values between the two roots is the set of values that satisfies the inequality $a x^{2} + b x + c < 0$ .
These are the values of $x$ for when the graph is below the $x$ -axis.
True or False?
If $x^{2} \leq n$ , where $n$ is a positive number, then $x \leq \pm \sqrt{n}$ .
False.
If $x^{2} \leq n$ , where $n$ is a positive number, then $- \sqrt{n} \leq x \leq \sqrt{n}$ .
The graph of $y = x^{2} - 7 x + 10$ is shown below.
Use the graph to find the answer to $x^{2} - 7 x + 10 > 0$ .
Shade the region which satisfies $x^{2} - 7 x + 10 > 0$ (above the x-axis).
The answer is $x > 5$ or $x < 2$ .
The graph of $y = x^{2} - 2 x - 15$ is shown below.
Use the graph to find the answer to $x^{2} - 2 x - 15 < 0$ .
Shade the region which satisfies $x^{2} - 2 x - 15 < 0$ (below the x-axis).
The answer is $- 3 < x < 5$ .
Outline how to solve a quadratic inequality, such as $a x^{2} + b x + c > 0$ .
To solve a quadratic inequality such as $a x^{2} + b x + c > 0$ :
- Find the roots of the quadratic equation $a x^{2} + b x + c = 0$ .
- Sketch a graph of the quadratic and label the roots.
- Identify the region that satisfies the inequality.
  For $a x^{2} + b x + c > 0$ the region above the x-axis satisfies the inequality.
State the formula for the discriminant of the quadratic function $a x^{2} + b x + c$ .
The formula for the discriminant of the quadratic function $a x^{2} + b x + c$ is $Δ = b^{2} - 4 a c$ .
This is given in the exam formula booklet.
What information is given by the discriminant of a quadratic function?
The discriminant of a quadratic function identifies the number of distinct real roots that its graph has. Also, it it identifies the number of distinct real solutions of the equation formed by setting the quadratic equal to zero.
How many real solutions does a quadratic equation have if the discriminant of the quadratic function is negative?
If the discriminant of a quadratic function is negative, then the equation has no real solutions.
How many real solutions does a quadratic equation have if the discriminant of the quadratic function is zero?
If the discriminant of a quadratic function is zero, then the equation has one (repeated) real solution.
If $a x^{2} + b x + c = 0$ has two distinct real solutions, what do you know about the discriminant of the the quadratic expression?
If $a x^{2} + b x + c = 0$ has two distinct real solutions, then the discriminant of the quadratic expression is positive.
True or False?
If the $x$ -axis is a tangent to the graph of the function $f (x) = a x^{2} + b x + c$ , then $b^{2} - 4 a c = 0$ .
True.
If the $x$ -axis is a tangent to the graph of the function $f (x) = a x^{2} + b x + c$ , then $b^{2} - 4 a c = 0$ .
If the graph touches the $x$ -axis once, then there is one repeated root to the equation $a x^{2} + b x + c = 0$ .
How many real roots does a quadratic graph have if the discriminant of the quadratic function is positive?
If the discriminant of a quadratic function is positive, then its graph has two distinct real roots.
The graph shows a quadratic function. What can you deduce about its discriminant?
The graph does not touch the $x$ -axis. Therefore there are no real roots, so the discriminant must be negative.

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