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A short class with Sarah and her students. The other ways are under More.
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Today, we're going to learn about the graph of a quadratic function, which we call a parabola. Can anyone tell me what they already know about parabolas?
I think they curve upwards or downwards, depending on something called 'a'?
Exactly! The direction of opening is determined by the coefficient 'a'. If 'a' is positive, the parabola opens upwards; if it's negative, it opens downwards. Let's remember this with the acronym 'OPEN': O for upwards, P for positive 'a', N for negative 'a', and E for 'equation of the parabola'.
So, if 'a' is zero, would it still be a parabola?
Good question! 'a' must never be zero for it to be a quadratic function. If it were, we'd have a linear function instead.
Now, let’s talk about the vertex, the most critical point of the parabola. The vertex can either be the highest point or the lowest point of our parabola based on the value of 'a'. Can someone remind me how we find the vertex?
Is it that formula, ?
That's right! And from that x-coordinate, we can substitute it back into the function to find the y-coordinate. The axis of symmetry also passes through this vertex.
So, the axis is just a vertical line? How do we write that?
Yes! To write it, we simply use the equation . Let's remember this with the mnemonic: 'AXIS = ALIGNED with X.'
Next, let’s identify the x- and y-intercepts of the parabola. The y-intercept is where the graph crosses the y-axis. How do we find it?
By setting 'x' to zero in the equation, which gives you 'c'.
Correct! And for x-intercepts, what methods can we use to solve for these points?
We can use factoring or the quadratic formula.
Completing the square also works, right?
Absolutely! All three methods are valid. Remember, the quadratic formula can also give hints about the number of real roots through the discriminant. Can anyone recall what the discriminant is?
It's !
Great! And the sign of the discriminant tells us: Positive means two real roots, zero means one real root, and negative means no real roots.
A polynomial function in standard form .
The graphical representation of a quadratic function.
The maximum or minimum point on a parabola, calculated by .
A vertical line through the vertex given by .
The value of the function when , which is .
Points where the parabola crosses the x-axis, determined by solving .
indicates the nature of the roots of the quadratic equation.
To find the vertex, don't you fret, just use , you won't forget!
Imagine a garden shaped like a U, the bottom is the vertex that you must view. Up or down it may stand, depending on 'a', that’s how you can understand.
Use the acronym 'VAX-YI' for Vertex, Axis, Y-Intercept and X-Intercept, they help find your way.
PARABOLA: P for Points, A for Axis, R for Roots, A for Area, B for Bottom, O for Orientation, L for Level, A for Axis of symmetry.
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What is a parabola?
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How do we find the vertex?
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What does the discriminant tell us?
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What is the axis of symmetry?
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A polynomial function of degree 2 in the form , where .
The U-shaped graph of a quadratic function.
The highest or lowest point on the parabola, found using .
A vertical line that divides the parabola into two mirror-image halves, represented by .
The point where the graph crosses the y-axis, calculated as .
Points where the graph intersects the x-axis, found by solving .
The expression used to determine the nature of the roots of a quadratic equation.
This section covers the graphical representation of quadratic functions, discussing key features such as the vertex, axis of symmetry, and types of intercepts.
In this section, we delve into the graph of a quadratic function, known as a parabola. We explore essential characteristics including how to identify its vertex, axis of symmetry, and intercepts. Practicing different methods of solving quadratic equations solidifies the understanding of these concepts.
In the study of quadratic functions, understanding the graphical representation is crucial. A quadratic function takes the form:
where . The corresponding graph is known as a parabola. The orientation of the parabola depends on the coefficient : it opens upwards if and downwards if .
Vertex: The vertex is the highest or lowest point on the parabola, representing the maximum or minimum value of the function. The coordinates of the vertex can be derived using the vertex formula:
Axis of Symmetry: This is a vertical line that divides the parabola into two mirror-image halves. Its equation is given as .
Y-Intercept: The point at which the parabola crosses the y-axis, found by evaluating .
X-Intercepts / Roots: The points at which the parabola intersects the x-axis, determined by solving . This can be achieved through factoring, using the quadratic formula, or completing the square.
Understanding the graph of a quadratic function enables us to apply its principles to various real-life scenarios, such as analyzing projectile motion or optimizing profits in economics.