( 4 parts 2 question) How many zeros does a quadratic function have? How can you determine the number of zeros

Question

a) from the graph?
b) from the factored form?
c) from the vertex form?
d) from the standard form?

Any help would be appreciated greatly, I've been trying to understand my math for a few days now, and I do not know how to answer this question

Answer 1

a) It has two zeros if the graph cuts the x axis in two points, one if it touches at 1 point (i.e. the vertex is on the x axis), and none if it fails to meet the x axis. This last case happens when the vertex is above the x axis and the graph is concave upwards, or vertex below and graph concave downwards.

b) If it has real factors, then it has two zeros if the factors are different from each other, one if they are equal, e.g.
(2x + 9)(2x + 9) = 0 has just one zero, -4.5

c)I'm not sure what you mean by the "vertex form". If it means

a(x - h)^2 + k, then it has two zeros if a and k have opposite signs, one if k = 0, and none if a, k have the same sign. e.g.

3x^2 + 30x + 17 can be written as

3(x + 5)^2 - 58, and since a = 3, k = -58 have opposite signs there are two zeros.

d) I assume "standard form" is what I call "general form", i.e.
ax^2 + bx + c.

In that case you just need to work out the discriminant, which is
b^2 - 4ac. If it's positive there are two zeros, if it's negative there are none, and if the discriminant is zero then the function has just one zero.

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Answer 2

A quadratic function has 2 zeros if you allow complex numbers. Restricted to real numbers it can have 2, 1, or 0.

From the graph: you see how many times it intercepts the x axis (the line y = 0)

From factored form: it is 0 whenever a factor is 0, so every linear factor provides a 0

vertex form: I am not sure what this is in general.

Standard form: Is this ax^2 + bx + cy^2 + dy + exy + f? If so there is no simple way to find the number of 0's, though by using calculus to find the mins and maxs, and trying a few test points you can do so.

If you can find any one point x0 where the value is negative and any other point x1 where the value is positive, then there are always 2 zeros for a quadratic function.

Answer 3

Zeros are numbers that make the value of the function zero. A quadratic can have zero, one or two zeros.

a) See where the graph crosses the x-axis. These are the zeros. If the vertex is above the x-axis, it has no real zeros. If it just touches the x-axis, it has one, and if it's below the x-axis, it has two.

b) Factored form: y=(x-a)(x-b). a and b are the zeros. If a and b are both imaginary, the function has no real zeros. If they are the same, it has one zero. If they are real and different, it has two zeros.

c), d) You take it from here. I think you have the idea.

Answer 4

The zeroes are the points where the graph crosses the X axis.
There can be no more than 2 (fundamental theorem of algebra),
but there can be zero (imaginary roots) or 1.
a. look at the graph. How many points touch the x axis.
b. are the factors real or imaginary or both
c. is there a vertex at y=0
d. is the value under the radical sign positive, negative or zero.

Answer 5

From the graph; wherever the function crosses the x-axis is a zero; however, you will not explicitly see imaginary zeros. The factors will reveal all zeros: the value of x that makes any factor zero will be a zero of the function. A quadratic function has two zeros (although they can be the same if the factors are the same).

Answer 6

ax^2 +bx +c is the final variety for quadratic equation. if b^2 - 4ac >0, it passes in the process the x-axis at 2 element b^2 - 4ac = 0, it passes in the process the x axis at one element b^2 - 4ac < 0, it would not pass in the process the x-axis.

Answer 7

from thegraph and/or the factoed form