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Graph Quadratic Functions in Standard Form 1Graph Quadratic Functions in Standard Form 1
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Question 1 of 4
1. Question
Graph `y=3x^2+6x`.Hint
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Standard Form of a Parabola
$$\color{green}{y}=a \color{green}{x}^{2}+b\color{green}{x}+c$$The value of `a` is positive, so the parabola is concave up. Solve for the `y`-intercept by substituting `x=0` into the equation.`y` `=` $$3\color{green}{x}^{2}+6\color {green}{x}$$ `=` $$3 ( \color{green}{0}^{2})+6(\color {green}{0}) $$ Substitute `x=0` `y` `=` `0` Mark the `y`-intercept on the graph.
Next, solve for the `x`-intercepts by substituting `y=0`.`y` `=` $$3\color{green}{x}^{2}+6\color {green}{x}$$ `0` `=` $$3\color{green}{x}^{2}+6\color {green}{x}$$ Substitute `y=0` `0` `=` `3x(x+2)` Factor out `3` `3x(x+2)` `=` `0` `3x` `=` `0` Equate factors to `0` `x` `=` `0` `x+2` `=` `0` Equate factors to `0` `x` `=` `-2` Mark the `x`-intercepts on the graph.
Draw a parabola using the points.
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Question 2 of 4
2. Question
Graph `y=8x-2x^2`.Hint
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Standard Form of a Parabola
$$\color{green}{y}=a \color{green}{x}^{2}+b\color{green}{x}+c$$Rewrite the equation so that it is in standard form.`y` `=` `8x-2x^2` `y` `=` `-2x^2+8x` The value of `a` is negative, so the graph is concave down.Find the `y`-intercept of the equation by substituting `x=0`.`y` `=` $$-2\color{green}{x}^{2}+8\color {green}{x}$$ `=` $$-2 ( \color{green}{0}^{2})+8(\color {green}{0}) $$ Substitute `x=0` `y` `=` `0` Next, solve for the `x`-intercepts by substituting `y=0`.`y` `=` $$-2\color{green}{x}^{2}+8\color {green}{x}$$ `0` `=` $$-2\color{green}{x}^{2}+8\color {green}{x}$$ Substitute `y=0` `0` `=` `-2x(x-4)` Factor out `-2x` `-2x(x-4)` `=` `0` `-2x` `=` `0` Equate factors to `0` `x` `=` `0` `x-4` `=` `0` Equate factors to `0` `x` `=` `4` Mark the `x`-intercepts on the graph.
Find the vertex from the formula `x=-b/(2a)``x` `=` `-b/(2a)` `=` `-8/(2(-2))` `a=-2`,`b=8` `=` `-8/(-4)` `x` `=` `2` Solve for the `y`-intercept of the vertex using the obtained value of `x`.`y` `=` `-2x^2+8x` `=` `-2(2^2)+8(2)` `=` `-2(4)+16` `=` `-8+16` `y` `=` `8` Draw a parabola using the points, together with the obtained vertex `(2,8)`
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Question 3 of 4
3. Question
Graph `y=2x^2-4x+3`.Hint
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Standard Form of a Parabola
$$\color{green}{y}=a \color{green}{x}^{2}+b\color{green}{x}+c$$The value of `a` is positive, so the graph is concave up.Find the vertex of the parabola by first solving for `x` from the formula `x=-b/(2a)`.`x` `=` `-b/(2a)` `=` `-(-4)/(2(2))` `a=2`,`b=-4` `=` `4/4` `x` `=` `1` Substitute the value of `x` into the quadratic equation.`y` `=` $$2\color{green}{x}^{2}-4\color {green}{x}+3$$ `=` $$2(\color{green}{1}^{2})-4(\color {green}{1})+3$$ Substitute `x=1` `=` `2(1)-4+3` `=` `2-1` `y` `=` `1` Simplify This corresponds to a vertex of `(1,1)`. Mark this on the graph.
Find the `y`-intercept of the graph. This is equal to `c`, which is equal to `3`. Plot this on the graph as well.
Connect the points and take note that the parabola opens up.
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Question 4 of 4
4. Question
Graph `y=-x^2+4x+5`.Hint
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Standard Form of a Parabola
$$\color{green}{y}=a \color{green}{x}^{2}+b\color{green}{x}+c$$The value of `a` is negative, so the graph is concave down.Find the `y`-intercept of the graph. This is equal to `c`, which is equal to `5`. Plot this on the graph.
Find the `x`-intercept of the graph by substituting `y=0`.`y` `=` `-x^2+4x+5` `0` `=` `-x^2+4x+5` `y=0` Since the equation is in standard form `(``a``x^2+``b``x+``c``=0)` we can factorise using the cross method.`-``x^2+` `4``x+` `5``=0`To factorise, we need to find two numbers that add to `4` and multiply to `5`
Read across to get the factors.`(-x+5)(x+1)`Solve for the `x`-intercepts and plot them on the graph.`-x+5` `=` `0` Solve for `x` `x` `=` `5` `x+1` `=` `0` Solve for `x` `x` `=` `-1` Connect the points and take note that the parabola opens up.
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- Sum & Product of Roots 1
- Sum & Product of Roots 2
- Sum & Product of Roots 3
- Sum & Product of Roots 4
- Solving Equations by Factoring 1
- Solving Equations Using the Quadratic Formula
- Completing the Square 1
- Completing the Square 2
- Intro to Quadratic Functions (Parabolas) 1
- Intro to Quadratic Functions (Parabolas) 2
- Intro to Quadratic Functions (Parabolas) 3
- Graph Quadratic Functions in Standard Form 1
- Graph Quadratic Functions in Standard Form 2
- Graph Quadratic Functions by Completing the Square
- Graph Quadratic Functions in Vertex Form
- Write a Quadratic Equation from the Graph
- Write a Quadratic Equation Given the Vertex and Another Point
- Quadratic Inequalities 1
- Quadratic Inequalities 2
- Quadratics Word Problems 1
- Quadratics Word Problems 2
- Quadratic Identities
- Graphing Quadratics Using the Discriminant
- Positive and Negative Definite
- Applications of the Discriminant 1
- Applications of the Discriminant 2
- Solving Reducible Equations